'kW'^-O^^-Cix. .V\ 082 IQIO A — A^ — ■- ■ o Aa =^ m 1 ^ 01 ^ ^S = CD 5 m - O . z = ^=? 7 1 3) 1 / = =^== 33 INTRODUCTION TO INFINITE SERIES OSGOOD SCENCE&ENGINEEBING UBBABV AUG 2 7 1997 Ews coaecTioN UCLA s- n, sJuZ-/n-<\c %au /^ I 9 INTRODUCTION TO INFINITE SERIES BY WILLIAM F. OSGOOD, Ph.D., LL.D. PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY XHIRID EDITION SCIENCE & ENGINEERING LIBRARY AUG 2 7 1997 EMS COLLECTION UCLA CAMBRIDGE publisbcCt bY> l3arvar^ *Cluivev5it\? 1910 1 • •( Copyright, 1897, by Harvard University. First edition, April, 1897. Reprinted with corrections, September, 1902. Reprinted, January, 1910. PREFACE. TN an introductory course on the Differential and Integral Calculus the subject of Infinite Series forms an important topic. The presentation of this subject should have in view first to make tlie beginner acquainted with the nature and use of infinite series and secondly to introduce him to thi' theory of these series in such a way that he sees at each step precisely what the question at issue is and never enters on the proof of a theorem till he feels that the theorem actually requires proof. Aids to tlu' attainment of these ends are : (a) a variety of illustrations, taken from the cases that actually arise in practice, of the application of series to computation both in pure and applied mathematics ; (b) a full and careful exposition of the meaning and scope of the more ditlicult theorems ; (c) the use of diagrams and graphical illustrations in the proofs. The pamphlet that follows is designed to give a presentation of the kind here indicated. The references are to Byerly's Differential Calculus, Integral Calculus, and Problems in Differential Calculus; and to B. O. Peirce's Short Table of Integrals; all published by Ginn & Co., Boston. WM. F. OSGOOD. Cambridge, April 1897. (/S2^ '■y introductio:n". 1. Examj^le. — Consider the successive values of the variable s,. r= 1 4-r+r2+ _^,.n-: for w = 1, 2, 3, Let r have the value ^. Then s,= l =i s, = 1 + -^ =1^ S3 = 1 + J + i =1^ ■ ••••• If the values be represented by points on a line, it is easy to see the S, = I S^ S, 3^2. — H 1 1 — r i ll Fiu. 1. law by which any s„ can be obtained from its predecessor, s„_i, namely : .s„ lies half way between s„_i and 2. Hence it appears that wlien )i increases without limit, Lim .s„ = 2. The same result could have been obtained arithmetically from the formula for the sum s„ of the first n terms of the geometric series a-\- ar-\- ar^ -\- + ar"-\ a(l— r") '" - 1 — 7- Here a =; 1, r = i, 1-i 2" , 1 = 2 — " 1 9'i — 1 ^ ~2 When /( increases without limit, ^^i^^i approaches as its limit, and hence as before Lim s„ == 2. 2 INTRODUCTION. § 2. 2. Definition of an Infinite Series. Let ? s„ , w' > ?t ; but that 2) always remains less than some definite fixed number, A: s„ < A for all values ofn, then s„ apjrroaches a limit, U: Lim .s„ =1 U. n = 00 §§ 4, 5. CONVEUGEXCE. 5 This limit, U, is not greater than A : U^- 6. A New Test-Series. It has just been seen that the series 1 + ^ + ^. + ^+ (3) converges when the constant quantity p ^ 2. We will now prove that it also converges whenever p ^ 1. The truth of the following inequalities is at once evident : - + -<- = — 2? ' 3? ^ 2^ 2^~ 1 I Kp I fip I 7/> ^^ /i/' AP-1 4P I 5? ' gp I 7/' ^ 4/- 4, i+i+ +i — 1 ^ 0, r <^ 1 and the series converges toward the hmit . Consequently no matter how many terms of the series i + l + i+ be taken, their sum will always be less than , and this series is therefore convergent, by the principle of § 4. Series (3) is useful as a test-series, for many series that could not be shown to be convergent by the aid of the geometric series, can be so shown by reference to it. For example. 7. Divergent Series. The series (3) has been proved convergent for every value of j9 ^ 1. Thus the series 1 + -^ + ^+^-+ 2 'V 2 3-^3 A'^ 4: is a convergent series, for j7 := 1.01. Now consider what the nu- morioal values of these roots in the denominators are : '7 2=1.007, '7 3 = 1.011, '7 4=1.014. In fact '7 100= 1.047 and '7 1000= 1.071; that is, when a thousand terms of the series have been taken, the denominator of the last term is multiplied by a number so slightly different from 1 that the first significant figure of the decimal part appears only in the second place. And wlu'u one considers that these same relations will be still more strongly marked when j) is set equal to 1.001 or 1.0001, one may well ask whether the series obtained by putting jj =^ 1, ' + l + l + \+ w is not also convergent. 8 CONVERGENCE. §§ 7, 8. This is however not the case. For 1,1, J 1 ^ 1 1 ^ n -\- n ^ 2 n 2 ' since each of the n terms, save the last, is greater than 1/2 n. Hence we can strike in in the series anywhere, add a definite number of terms together and thus get a sum greater than h, and we can do this as often as we please. For example, 1 I 1 ^ 1 '' = '^ 3 + 4>2 ''=^^ 5 + 6 + 7 + 8>2 " = ^' 9 + To + + 1-6 > 2 Hence the sum of the first n terms increases without limit as n increases without limit, or Lim .S,, rrr 30 The series (4) is called the harmonic series. How is the apparently sudden change from convergence for p ^ 1 in series (3) to divergence when p = 1 to be accounted for? The explanation is simple. When p is only slightly greater than 1, series (3) indeed converges still, but it converges towards a large value, and this value, which is of coui'se a function of p, increases without limit when j>, decreasing, approaches 1. When p = 1, no limit exists, and the series is divergent. 8. Test for Divergence. Exercise. Establish the test for diver- gence of a series corresponding to the test of §5 for convergence, namely : Let ^'o + ^'i + (a) he a series of positive terms that is to be tested for divergence. If a sev'es of positive terms already knoiun to be divergent "0 + «i + (/8) can be found vjhose terras are never greater than the corresp)onding terms in the series to be tested (a), then (a) is a divergent series. Examjjles. §§ 8, 9. CONVEUGENCE. 9 14-l-f- + -+ p 1, it is divergent, if T := 1 there is no test : Lim ^K+i _ ^ ^ 1 Convergent; n " T > 1 , Divergent ; (( T = 1, No Test. First, let T <; 1. Then as n increases, the points corresponding to the values of ii„ + ^/u„ will cluster about the point t, and hence if r y I — I \ 1 Y— Vu;. 4. a fixed point y be chosen at pleasure between t and 1, the points w« + i/'^i ""'ill^ fo^' sutiiciently large values of n, i.e. for all values of n equal to or greater than a certain fixed number m, lie to the left of y, and we shall have 10 CONVERGENCE. § 9. x<. w ^ m ; or, n = m, X < ^• ^™+i < ^^y» n =1 m -{- 1, %i + 1 M„,+2 < w„,+iy < w„,y', n =z 711 -\- 2, w„+3< w« + 2y < «„.y', < ^^„ (y + y' + y' + + yO < ^„ p Adding p of these inequalities, we get J^ — y Tlie sum of the terms ?«, beginning with ?<,„ + i never, therefore, y rises as high as the value u^ — - — • Hence the w-series converges. 1 — y The case that t ^ 1 ( or t := oc ) is treated in a similar manner, and may be left as an exercise for the student. If T = 1 there is no test. For consider series (3). The test- ratio is «„ (»^ + 1)'' V ''. and hence t ^ 1 , no matter what value p may have. But whenp ^ 1, (3) converges; and when i> <^ 1, (3) diverges. Thus it appears that T can equal 1 lioth for convergent and for divergent series. Memark. The student will observe that the theorem does not say that the series will converge if u„^i/u,^ becomes and remains less than 1 when n increases, but that it demands that the limit of u„ + i/u,^ shall be less than 1. Thus in the case of the harmonic series this ratio n/(n -\- 1) is less than 1 for all values of n, and yet the series diverges. But the bmit is not less than 1. Examples. Are the following series convergent or divergent? 11 21 2 3 3~'3'5''3'5'7~^ 1 + 1 + 1 + 1+. . 9 92 93 1- + — 4-1 9100 I glOO 1 ^lOi 2 ! 3 ! 4! _ -(- -:i^ 4- -11. + 100 ' 100^ ' 100^ ' • §§10,11. CONVERGENCE. 11 10. A Further Test-Ratio Test. The following test for conver- gence and divergence is sometimes useful ; the proof of the rule is omitted. If approaches a limit, let this limit be denoted by a-. Then the series Uo + Wi -f converges, if o" ^ 1 (o^ if a- = oo) ; diverges, if a <^ 1 (or if a- = — oo) ; if o- = 1 , there is no test. Example. 1-2 ^ 3-4 ^ nf l_!l^^A- >.^ (2n — l)2n ^_ 4 -f i ^n-fA_ /^ (2n — l)2n \ ''n J V (2// +l)(2n + 2)y' 2 + ^+(;-)^ and o- = 2 ; the series converges. Test the following series : 11-3 1-3-5 2 '~ 2^ ' 2'4-6 ' ^-^m^im^ 22 — a ~ 32 — a ' 4''^ — a ^ Apply any of the foregoing tests to determine the convergence or the divergence of the series on pp. 45, 46 of Byerly's Problems in Differential Calculus. b) SERIES WITH BOTH POSITIVE AND NEGATIVE TERMS. 11. Alternating Series. Theorem. Let the terms of the given series \) he alternately x>ositive and negative: «o — "1 + «j — ''3 + ; (5) 2) let each u he less than (or equal to) its predecessor : w„ + i<«„; 8) let Lim »,, = 0. n ^ cc Then the series is convergent. 12 CONVERGENCE. § 11. The following series may serve as an example. 2^3 4 1.1 1 , 1-9 + 5^-7+ (6) Proof. Let \ = ^h — '^'l-\-^h + (— ly-^u^-i and plot the points Si, Sg, Sg, Then we shall show that the points Si, S3, S5, s-2m + \i always move to the left, s^ s^ Se Ua u; S5 S3 s, —\ \ — \M \^ — \ \ — Fig. 5. but never advance so far to the left as So, for example. Hence by the principle of § 4 they approach a limit, Ui : TO ^ 00 Similarly, the points s,^ ^ii H^ ^-zmi always move to the right, but never advance so far to the right as s^, for example ; hence by the same principle they also approach a limit, U2 : Lim .S2,, z= U^ . TO = 00 Finally, since lim S2„, + i = lim .s\,,„ + lim ru^ ; TO ^ 00 m ^ -«2 1111(1 -v. < Si. For S2„. + i = .Sj,,, -\- «2,„ > «2 + M2». > «2» «2m == ■''2m M "2,,. ^ ^1 "2,,, + +^7+^>2 and does not converge toward as its limit. This fact affords a now proof of the divergence of the harmonic series. It may be remarked that the more general condition Lim [;/,, + »„ + i + + "„ + ,.-i] = 0> where j> may vary with n in nuy icise ice choose, is a sutlicient con- dition for the convergence of the series. See Appendix. 14. Convergence. The General Case. Let be an}^ series and let -^'o + '"i + denote the series of positive terms, — ?(\, — >r, — the series of negative terms, taken respectively in the order in which they occur in (a). For example, if the »-series is i_i+i_i+ 2 ' 2- 2* ' l(i CONVERGENCE. § 14. then the ti-series is 1 -I- 22 I 2^ 1 + ^. + ^4 + and the — to-series is _1_1_1_ 2 2^ 2^ Let s^= Uo-\- Vi-\- -\- w„_i, «■,«= ^'o + ''i + + %n-i^ ^ = ic^ -\- IVi -\- + ?«^_i, T Then, whatever value n may have, 5^ can be written in the form •5« = <^,n — '^p- Here m denotes the number of positive terms in s , o- their sum, etc. When n increases without limit, both m and p increase without limit, and two cases can arise. Case I. Both a^^^ and t^, approach limits : Lim o-„^ = V, Lim t^, =: W; m ^ OO jp :^ 00 so that both the r-series and the t«-series are convergent. Hence the u-series will also converge, Lim 8^ = U, n ^ 1 or ^ < — 1 , then t > 1 u' and "l*"^ "> 1, when w > m. §§ 16, 17. CONVERGENCE. 19 Hence w'„.^i > m',„, ^^'-. + 2> "',,+ 1 >^^'„., <,. + 3 >«'„, + 2 >m'„, or u'„ > «'„,, n > m ; that is, all the u'^'s from n := m on are greater than a certain positive quantity p = w',,^ and hence u\^ and ?<^ cannot approach as their limit, when n ^ co. Example. In the series of § 15, < =r — x-; hence this series di- verges for all values of x numerically greater than 1. These results may be represented graphically as follows : — Divergent — 1 1 Divergent Conrergmt Exercise. For what values of x are the following series conver- gent, for what values divergent ? Indicate these values by a diagram similar to the one above. x"^ x^ Ans. — 1 ^ a; <^ 1 , Conv. ; x^\^x<^ — 1 , Div. x^ , x^ x'' , ^-3 + 5-7 + l+x- + ^ + 3^+ 10a; + 102.1-2 + 10«.»-3 -|- 1 + .r -|- 2 ! a-2 -|- 3 ! x^ -\- 17. Theorem. Let «0 + «1 + «2 H- be any ahsoluteUj convergent series; po^ pi- p>i (^^^y set of quantities not increasing numerically indejinitely. Then the series converges absolutely. For, let a',,, p\^ be tlie aiisolute vahics of k^^, p_^ respectively, fl a positive quantity greatt'r than any of the q^iantities p\^, and form the series «'op'o + <'\p\ + ^f'.p'. -h 20 CONVERGENCE. §§ 17, 18. The terms of this series are less respectively than the terms of the convergent series and each series is made up exclusively of positive terms. Hence the first series converges and the series converges absolutely. Examples. 1. The series sin a; sinS.i' sin 5 a; 12 g2 "I T2 converges absolutely for all values of x. For the series converges absolutely and sin nx never exceeds unity nmnerically. 2. If Oq + «! + «o + and &i + &2 + are any two absolutely convergent series, the series f'o ~\- f^i cos X -\- tto cos 2.1; 4- and bi sin x -(- 63 sin 2 a; -|- converge absolutely. 3. Show that the series e~^ cos X -|- e^^^ cos 2x -\- converges absolutely for all positive values of x. 4. What can j'ou say about the convergence of the series 1 + r cos ^ -|- r^ cos 2 ^ + ? 18. Convergence and Divergence of Power Series. A series of ascending integral powers of a,', a,, -\- a^x -\- a.x- -f- , where the coefficients Oq, «i, «25 ^^'^ independent of x, is called a jwiver series. Such a series may converge for all values of X, but it will in general converge for some values and diverge for others. In the latter case the interval of convergence extends equal distances in each direction from the point x = 0, and the series con- Divergent — r r Divergent Convergent verges absolutely for every point x lying ivithin this interval, but not necessarily for the extremities of the interval. § is. CONVERGENCE. 21 Tlie proof is as follows. Let Xq be any value of x for which the terms of the power series a^x^ do not increase without limit ; a'^, x'^, the absolute values respectively of a„, x„. Then a'^x'," is less than some fixed positive quantity C, independent of j(, for all values of n. For X =^ a'o, the power series may converge and it may diverge. — Let h be any value of x numerically less than x'q ; h' its numerical vahic. Then the power series converges absolutely for x = h. For a\^''^ = a'^xV C'yX < Cr" , wdiere r = Ji'/x'q <^ 1. Hence the t(Mnis of the absolute value series a'oH- a\h' -\- — 4 . 3 - 3 ? and then take the logarithm of each side : log 2 = log A + log I = .287 (;-)) -f- .405 (5) = .693 (O); Hence, to three places, log 2 =^ .693. Next, to find log 5. Here the series must be applied in still a different way, for if 1 -{- h be set equal to 5, /i == 4, and the series does not converge. We therefore set 5 = 4+ 1 =4(1 +i), log 5 =r 2 log 2 -|- log IJ =: 1.386 (0) + .223 (2) = 1.609 (2), where log 1^ is computed directly from formula (8). From the values of log 2 and log 5, log 10 can at once be found. log 10 = log 2 4- log 5 = .693 (0) -f 1.609 (2) = 2.302 (2) or to 3 places. This latter logarithm is of great importance, for its value must be known in order to compute the denary logaritlim from the natural * The formula is nevertheless useful as showing the value of a familiar series, (0). We could not find by direot conipntation the value of this series to, say, seven places, because the work would be too long. log 10 = 2.302. 24 SERIES AS A MEANS OF COMPUTATION. §§ 20, 21. logarithm. By the formula for the transformation of logarithms from the base c to the base ?>, , . log, J. we have ^^S- ^ = log:iO ' Hence for example .693 ">«-' = 2m = ■'''■ Examples. Compute log 20, logio20, log 9, logio9, log 13, logiol3. . ■ 21. Series (8) is thus seen to serve its i>uipose well when only a few places of decimals are needed. Suppose however we wished to know log 2 correct to 7 places of decimals. Series (8) would then give less satisfactory results. In fact, it would require 16 terms of the series to yield log li to 7 places. From (8) a new series can be deduced as follows. Let h z= — x. Then (8) becomes log (l—x) = —x—- — ~— Next replace h in (8) by x : x'- , x^ log (1 +.«) = + X — - + - — Subtracting the former of these series from the latter and combining the logarithms we get the desired formula : iJi5= , (, + 1 + ^' We have subtracted on the right hand side as if we had sums. We have not ; we have limits of sums. This step will be justified in §35. "We will now apply series (9) to the determination of log 2 to seven places. X must be so chosen that ~ — — ; =. 2, 1. e. ^ =^ ^ and , 1 + i /I , 1 1 , 1 1 , 1 1 , \ ^^^1-^=1=^3 + 3 3^ + 5 3^+7 3^+ ) § 21. SEIilKS AS A MHANS l)V C(XMl'LTATION. 25 The advantage of this series over (8) is twofold : first, it suflices to compute tlie value of the series for one value of .r, x = ^, and second, the series converges more rapidly than (H) for a given value of X, since only the odd powers of x enter. (i) = .333 333 33 (i) = .333 333 33 G)« = .037 037 04 1 7 ■ ar = .012 345 68 (hr = .004 115 23 1 ■ i-hr = .000 823 05 a)' - .000 457 25 1 T ■ ar = .000 065 32 ar = .000 050 81 1 ■g- ■ (hr = .000 005 65 (i)" = .000 005 65 1 11 •(^)" = .000 000 51 (^y = .000 000 G3 1 ■(iy' = .000 000 05 (h'' = .000 000 07 1 TZ ■(hy' = .000 000 00 .346 573 59 The term -j^ (^y^ has no effect on the eighth decimal place. But this is not enough to justify us in stopping here. We must show that the remainder of the series from this point on cannot influence this place either. Now the remainder is the series 15 3^6 ^ 17 3" ^ 19 318 ^ 15 315 L 17 32^ 19 3*^ J The value of the series in brackets cannot be readily determined ; nor is tliat necessary, for it is obviously less than the value of the series obtained from it ])y discarding the coefiicients ^, ^|, etc., i. e. than the geometric series , 1,1, 19 3'^ ' 3* ' 1 — I 8 and hence the remainder in question is less than J_ J^ 9 15 3^5 8' and so does not affect the eighth place. We obtain then finally for log 2 the value 2 X .346 573 5(9) =.693 147 1(8) or to seven places log 2 = .693 147 2. 26 SERIES AS A MEANS OF COMPUTATION. §§ 21, 22. Examples. Show that log li = .223 143 (4) log 5 = 1.609 437 (8). Compute log 2 by aid of the formula log 2 = — log ^ = — log I — log |. Knowing log 2 and log 5 we can find log 10 : log 10 = 2.302 585. Exam2)le. Compute logio 2 , logio 9 to six places. Series (9) is thus seen to be well adapted to the computation of logarithms. If y denote any positive number and x be so deter- mined that 1 + .^; . .V — 1 — ! — = y , I.e. X =1 — , — - , then X always lies between — 1 and -j- 1 and series (9) converges towards the value log y. For values of y numerically large the convergence will be less rapid and devices similar to those above explained must be used to get the required result. In the actual computation of a table, not all the values tabulated are computed directly from the series. A few values are computed in this way and the others are found by ingenious devices. b) THE BINOMIAL SERIES. 22. In elementary algebra the Binomial Theorem for a positive integral exponent : 7)z ( ')yh 1 1 (a + hy = a'" + ma'"-i6 -] ^—-^ — La'^-'^h^ -\- i. • ^ (to m -\- 1 terms) is established. Consider the series 1 -h /^^ + 1.2 ^ 1 • 2 • 3 ^ If /a is a positive integer, this series breaks off with /a + 1 terms, for then, from this point on, each numerator contains as a factor. Thus if /A r= 2, we have 2-1 2 • 1 • 1 _j_ 2x -^ x^ -\ — - x^ -f" 6tc. (subsequent terms all 0), § 'I'l. SERIES AS A MEANS OF COMPUTATION. 27 or simply 1 -\- 'Ix -\- x^. In this case the series is seen by compari- son with the binomial formula (a := I, h z= x, m z=z fj.) to have the value (I -\- x)f: \ I '' ^ ^ ^ 1 ■ 2 ' 1-2-.3 ' If however /a is any nuinlter not a positive inte 23. Series for sxn~'^h and tan~^li. Tlic Co mputation of n. The method set forth in § 19 is applicable to the representation of 8in~^/i and tan~^/i (v. Exercise, § 19) by series. Jr*h dx 1 /<3 1.3 7,6 tan-i/( = h— -\- y — (12) From these series the value of ir can be computed. If in series (12) wo set /; r= 1, we get the equation : 4 3^5 7 ^ This series, like series (6), is not well adapted to computation. A better series is obtained by putting h ^ ^ \\\ series (11) : 6 2 ' 2 ;} \'>) '2-4 f) \->) ^ This series yields readily three or four places of decimals; but if greater accuracy is desired, more elaborate methods are necessary, (v. Jordan, Cours d' Analyse^ Vol. I, § 2r)2 ; 1893). Exercise. If the radius of the Earth were exactly 4000 miles, to how many places of decimals should you need to know v in order to compute the circumference correct to one inch? Determine ir to this number of places by Jordan's method. 24. The Length of the Arc of an Ellipse. Let the equation of the ellipse be given in the form : X ^= a sin <^ , y = b cos <^ . Then the length of the arc, measured from the end of the minor axis, will be VI — e^ sin-^ dcji , s where (a^ — b'^)/a- = e^ <^ 1. The integral that here presents itself is known as an Elliptic Integral and its value cannot be found in the usual way, since the indefinite integral cannot be expressed in terms of the elementary functions. Its value can however be obtained by the aid of infinite series. The substitution of esin<^ for x in the last example of § 22 gives the formula V 1 — e^ sin^ = l — -e^ sin^ <^ — — — e* siu^ — 30 SERIES AS A MEANS OF COMPUTATION. §§ 24, 25. Hence (v. § 40) s=z aTcfi — ^e^ f'^sm^cj>dcf> — ^e* I sm'd 1 These integrals can be evaluated by the aid of the formulas of IV of Peirce's Short Table of Integrals. In particular, the length of a quadrant S will be found by putting (ji = ^ n and using the formula (No. 483 of the Tables, 1899, or later, edition) TT J~- 1 • 3 • 5 {n~ 1) TT sin" d) dA = - — - — ^ ^ , n, an even integer. 2-4-6 n 2 The elliptic integral then becomes the integral known as the Complete Elliptic Integral of the Second Kind ; it is denoted by E : TT E = j^ 1 — e^sin-^c^ r?<^ . (No. 248 of the Tables). - aE . If e =: the ellipse reduces to a circle and S = ^-na. Examples. 1. Compute the perimeter of an ellipse whose major axis is twice as long as the minor axis, correct to one tenth of one percent. 2. A tomato can from which the top and bottom have been removed is bent into the shape of an elliptic cylinder, one axis of which is twice as long as the other. Find what size to make the new top and bottom. If the original can held a quart, how much will the new can hold ? 25. The Period of Oscillation of a Pendulum. It is shown in Mechanics (v, Byerly's Int. Cal., Chap. XVI) that the time of a complete oscillation of a pendulum of length I is given by the formula \9 Jo V 1 — k''sm' sin-, where a denotes the initial inclination of the pendulum to the vertical. K is known as the Complete Elliptic Integral of the First Kind and its value is computed as follows. The substitution of Z:sin<^ for a; in the series for (1 — a;^)~i gives the formula (v. Exs., § 22). ^1 — k^.s'm'^; /, the error in the computed value of r is inversely proportional t(; tlie length of the column of mercury used, — a result not a priori obvious, for r itself is inversely propor- tioned only to V '• Exercise. An engineer surveys a field, using a chain that is incorrect by one tenth of one percent of its length. Show that the error thus arising in the determination of the area of the fieM will be two tenths of one percent of the area. 28. Pendulum Problems. A clock regulated Ity a pendulum is located at a point (A) on the earth's surface. If it is carried to a neighboring point (B), h feet above the level of (^-1), show that it will lose ^5^ h seconds a day, i. e. one second for every 244 feet of elevation. The number of seconds N that the clock registers in 24 hours is inversely proportional to the period T of the oscillation of the pen- dulum. Hence (cf. §25) N where the unpriraed letters refer to the location (A) , the primed letters to (B). If the clock was keeping true time at (A), then N =i 86,400. g- {R + hy' where R denotes the length of the radius of the earth. (Cf. Byerly's Diff. Cal., §117.) Hence A^— ^y := .V ( 1 — ]■- ] = N '^ Mh 7? + 7/ R R' ^ R^ If h does not exceed 4 miles, h/R < .001, h^/R- < .000 001, and the first term of the series gives N — N' correct to seconds : Examples. 1. The summit of Mt. AVashington is G226 feet above the sea level. How many seconds a day will a clock lose that keeps accurate time in Boston Harbor, if carried to the summit of the Mountain ? 2. A pendulum that beats seconds on the surface of the earth is observed to gain one second an hour when carried to the bottom of a mine. How deep is the mine? Assume the attraction at interior points of the earth to vary as the distance from the centre. 34 SERIES AS A MEANS OF COMPUTATION. § 29. 29. Exercises. 1. Show that the correction for expansion and contraction due to heat and cold is given by the formula 71 = 43,200 a ^, ■where a denotes the coefficient of linear expansion, t the rise in temperature, and w the number of seconds lost in a day. For brass, a =^ .000 019, t being measured in degrees centigrade. Thus for a brass pendulum » = .82 t, and a rise in temperature of 5° causes the clock to lose a little over 4 seconds a day. 2. A man is standing on the deck of a ship and his eyes are h ft. above the sea level. If D denotes the shortest distance of a ship away whose masts and rigging he can see, Ijut whose hull is invisible to him, hi the height, measured in feet, to which the hull rises out of the water, show that, if refraction can be neglected, D = 1.23 (V ^^ + V ^^i) miles. If // z= /^i = 16 ft., /> = 10 miles (nearly). 3. Show that an arc of a great circle of the earth, 2 J miles long, recedes 1 foot from its chord. 4. Assuming that the sun's parallax is 8". 76, prove that the dis- tance of the sun from the earth is about 94 million miles. 5. Show that in levelling the correction for the curvature of the earth is 8 in. for one mile. How much is it for two miles? 6. The weights of an astronomical clock exert, through faulty construction of the clock, a greater propelling force when the clock has just been w^ound up than when it has nearly run down, and thus increase the amplitude of the pendulum from 2° to 2° 4' mi each side of the vertical. Show that if the clock keeps correct time when it has nearly run down, it will lose at the i-ate of about .4 of a second a day when it has just been wound up. 7. Two nearly equal, but unknown resistances, A and B, form two arms of a Wheatstone's Bridge. A standard box of coils and a resistance x to be measured form the other two arms. A balance is obtained when the standard rheostat has a resistance of r ohms. When however A and B are interchanged, a balance is obtained when the resistance of the rheostat is r' ohms. Show that, ap- proximately, x= .H>-+ '•')• 8. The focal length /of a lens is given by the formula --- + - / 2h ^ P-2 ' § 29. SERIES AS A MEANS OF COMPUTATION. 35 where Pi and p, denote two conjugate focal distances. Obtain a simpler approximate formula for ./" that will answer when p, and jh are nearly equal. •). "A ranchman 6 feet 7 inches tall, standing on a level plain, agrees to buy at S7 an acre all the land in sight. How much nuist he pay? Given 640 acres make a square mile." Admission Exam, in Sol. Geom., June, 1895. Show that if the candidate had assumed the altitude of the zone in sight to be equal to the height of the ranchman's eyes above the ground and had made no other error in his solution, his answer would have been 4 cents too small. 10. Show that for small values of h the following equations are approximately correct {h may be either positive or negative) (1 -J- h)"' = 1 -1- mil . . Hence {1 -\- hy = I -\- 2Ji ; ^ 1 -\- h = 1 -\- ^h; 1 . _; 1_ 1 + /^- '' (1 + /0' ^ = 1 — f^h . V 1 -f /i If h, Ic, I, p, are all numerically small, then, approximately, (1 +/r) (1 + 70 (1 + /) = 1 +^' + ^- + /+ , (1 + (i-hp) ^ ^ = 1-/'; 7^-r-M2 = i-2'^; III. TAYLOR'S THEOREM. 30. It is not the object of this chapter to prove Taylor's Theorem, since this is done satisfactorily in any good treatise on the Differ- ential Calculus ; but to indicate its bearing on the subject under con- sideration and to point out a few of its most important applications. It is remarkable that this fundamental theorem in infinite series admits a simple and rigorous proof of an entirely elementary nature. Rolle's Theorem, on which Taylor's Theorem depends, and the Law of the Mean lie at the very foundation of the differential calculus. From Rolle's Theorem follows at once the theorem contained in the equation fix, + 70 =/(-^o) +/' (^o) A +/" (xo) ^ + • • • • +r" (a-o + eh) ^' , (13) < ^ < 1. This latter theorem is frequently refen'ed to as Taylor's Theorem with the Remainder i?„ =: /<"' (Xq -\- Oh) —^ • It includes the Law of the Mean /(o-o + ^0 — ./'(-^o) = Kf O^-o + Oh) (14) as a special case and thus affords a proof of that Law. If in (13), when n increases indefinitely, R^ converges towards as its limit, the series on the right hand side of (13) becomes an infinite power series, representing the function fix, -\- h) throughout a certain region about the point X(^ : fix, + h)=fix,) J^fix,)h+r{x,) 1^ + (15) This formula is known as Taylor's Theorem and the series as Taylor's Series. The value x, is an arbitrary value of x which, once chosen, is held fast. The variable x is then written as x, -\- h. The object of this is as follows. It is desired to obtain a simple representation of the function /(x) in terms of known elements, for the purpose of com- puting the value of the function or studying its properties. One of the simplest of such forms is a power series with known coefficients. §§80,31. tavi.ok's tiikouem. 37 Now it is usually impossible to represent /(a;) by one and the same power series for all values of a,', and oven when this is possible, the series will not converge rapidly enough for large values of the argu- meijt to 1)6 of use in computation. Consequently wo confine our attention to a limited domain of values, choose an Xq in the midst of this (loniaiii, and roplaco the independent variable x by A, where X = x^, -\- h, h = X Xq. The vahies of .)• for the domain in (luestion may not be small, but the vahies of // will l)c, // = corresponding to x =z Xq. If x^ is so chosen that /(Xq), /'(a'o), /"(a'o), ad in/, are all finite, then the value of f(x) for values of x near to .Tq, i. e. for values of h numerically small, will usually* be given by Taylor's Theorem. An example will aid in making clear the above general statements. Let f(x) = log X. Then it is at once clear that f(x) cannot be developed l)y Taylor's Theorem for Xq =z 0, for/(0) = log := — x . It is just at this point that the freedom that we have in the clioice of .fy stands us in good stead; for if we take x^ greater than 0, then /(a-'o), /' (x^), /"(Xq), will all be finite and /(.ro -|- /i) can be developed by Taylor's Theorem, the series converging for all values of h lying between x^ and — Xq. The proof is given for Xq =: 1 in the Dijf'. CuL, § 130. Thus we have a second proof of the development of log (1 + h), (formula (8) of § 19). 31. Ttco Applications of Tuijlofs Theorem with the Remainder, (13). This theorem, it will i)e observed, is not a theorem in infinite series. Any function whose first n derivatives are continuous can l)e expressed in the form (13), while the expression in the form (15) requires the proof of the possibility of passing to the limit when ?l zz= X . Thus (13) is a more general theorem than (15) and it avoids the necessity of a proof of convergence.! It is because of the applica- tions that (13) and (15) have in common, that it seemed desirable to treat some applications of (13) here. * * Exceptions to this rule, though possible, are extremely rare in ordinary practice. t It is desirable that (13) should be applied much more freely th;in has hitiierto been the custom in works on the Infinitesimal Calculus, both bi'cau 0. Then f(x, + h) = f(x,) + /(^") (x, + Oh) ^^ • The equation of the tangent is now 2/2 = f(Xo) and 2/1 — 2/2 = f'"' (^o + Oh) ^|^ f-^"^ (x) vn.\\ in general be continuous near the point x z= Xq and it is positive at this point ; it will therefore be positive in the neighborhood of this point and hence 2/i — 2/2 > both for positive and for negative values of h, i. e. the curve lies above its tangent and has therefore a minimum at the point x = Xq. Similarly it can be shown that if /(^'O (^o) < 0, all the earlier deri- vatives vanishing, f(x) has a maximum in the point a'o. Lastly, let f'(Xo)*= 0, P'-H^'o) = 0, /^^"^"(^•o) H= 0. * The student should illustrate each case in tliis § by a figure. §31. Taylor's THEOREM. 39 Then y, — y, = f^" + '^ (./•„ + 6h) ^'"'^' (2n-\- 1)! y(2,. + i)^^-j Avill ill general he conliiiuoiis near x := Xq and it will therefore preserve the same sign for small values of /i, positive or negative; but /i^" + ^ changes sign witii //. Hence the curve lies on opposite sides of its tangent on opposite sides of the point x^ and this is then a point of inflection. Exercises. 1 . Show that the condition for a point of inflection not parallel to the a'-axis is /"(.^•o) =: 0, r'"^(x,) z= 0, r'"^'^(xo) dp 0, / 2« + i) ^^^ being continuous iieur r = Xq. 2. Show that a perpendicular drawn to the tangent from a point P' infinitely near to a point of inflection P is an infinitesimal of higher order than the second. Curvature. The osculating circle was defined (Diff. Cal. § 90) as a circle tangent to the given curve at P and having its centre on the inner normal at a distance p (the radius of curvature) from P. We will now show that if a point P' be taken infinitely near to P and a perpendicular P'M be dropped from P on the tangent at P, cutting the osculating circle at P", then P'P" is in general an infinitesimal of the third order referred to the arc PP as principal infinitesimal. Let P be taken as the origin of coordinates, the tangent at P being the axis of x and the inner normal the axis of // ; and let the ordinate y be represented by the aid of (13). Here a-o=0, x=h, f(0)=f\0)=0, /"(0)>0, and y = ^f" (0) .r^ + }/'" (Ox) x'^. The radius of curvature at P is a p - DJ'y /"(O) and the equation of the osculating circle is ^''' + (.'/ — Pf = P- Hence the lesser ordinate y' of this circle is given by the formula X* y' = p - ^ p^ - x^ = p - p (1 - I -- l- -^^ ) * Instead of the infinite series, formula (13) might have been used here, with n = 4. But we happen to know in tliis case that the function can be developed by Taylor's Tlieorcni (15). .. 40 TAYLOR'S THEOREM. §§ 31, 32. and y-y'^ -c' (^if"'{ex) - i i -'^ ) • From this result follows that (y — y')/x^ approaches in general a finite limit different from 0, and hence that y — y' is an infini- tesimal of the third order, referred to P'M = a; as principal infini- tesimal. But P'M and PP' are of the same order. Hence the proposition. Exercise. Show that for any other tangent circle y — y' is an infinitesimal of the second order. Second Application : Error of Observation. Let x denote the magni- tude to be observed, ?/ = / (x) the magnitude to be computed from the observation. Then if .Jo be the true value of the obseived magni- tude, X ^ Xq -\- h the value determined by the observation, h will be the error in the observation, and the error H caused thereby in the result will be (c/, (14)) H = fix, + h) — /(a-o) = /' (X, + Oh) h. In general f'(x), will be a continuous function of x and thus the value of f(xo -\- 6h) will be but slightly changed if x^ -^ Oh is replaced by x. Hence, approximately, H = f'{x)h and this is the formula that gives the error in the result due to the error in the observation. 32. The Principal Apx>lications of Taylor's Theorem icithout the Remainder, i. e. Taylor's Series (15) consist in showing that the fundamental elementary functions: e"^, sina:;, cos.r, log.i;, x^, sin~^.^*, tan~^.c can be represented by a Taylor's Series, and in determining explicitly the coefficients in these series. It is shown in Ch. IX of the Diff. Col. that these developments are as follows.* e'-" x^ . x° 1 + -^^ + 9T + sT + ^.3 ^.5 sinxr. ...--+_, X^ , X* COS .T = 1 — - + 4 , These developments hold for all values of x. * The developments for sin~'x and tan- 'a; are to be sure obtained by in- tegration ; but the student will have no difficulty in obtaining them directly from Taylor's Theorem. §§ 32, 33. Taylor's theorem. 41 loga; =\og(l -\- h)= h — ~ -\- ^— x^^ = (!-{- hy z.. 1 + ;x/. + '"\~^^^ /^^ + ' 2 3 ' 2 • 4 .5 ' x^ , x-^ tan-i.t; = .x — - + - — These developments hold for all values of h (or, in the case of the last two formulas, of x) numerically less than 1. Exercise. Show that sin a; can be developed aliout anj' point x,, by Taylor's Theorem and that the series will converge for all values of h. Hence comi)ute sin 46° correct to eight places of decimals. 33. As soon however as we pass beyond the simple functions and try to apply Taylor's Theorem, we encounter a ditliculty that is usually insurmountable. In order namely to show that f{x) can be expanded by Taylor's Theorem it is necessary to investigate the general expression for the n-i\\ derivative, and this expression is usually extremely complicated. To avoid this difHculty recourse is had to more or less indirect methods of obtaining the expansion. For example, let it be required to evaluate X 1 ef — e~'' , ■ ax. X The indefinite integral cannot be obtained and thus we are driven to develop the integrand into a series and integrate term by term. Now if we try to apply Taylor's Tlieorem to the function {f — e~'')/x, the successive derivatives soon become complicated. We can how- ever proceed as follows : X- x^ e^ = 1 + .r + ., , -f .. , + , o—x X^ X^ = '-^+2-!-3! + e- — . .. , ^^ / .)■« x^ \ e- _, -X = 2 (^.r 4- _.,, + _.,+ j; and hence, dividing through by x, wo have 42 TAYLOR'S THEOREM. § 33. X^^'"-^=K'+3^ + ^+ ) = 2.114 502. Examples. Do the examples on p. 50 of the Problems. General Method for the Expansion of a Function. To develop a function /(•'c), made up in a simple manner out of the elementary functions, into a power series, the general method is the following. The fundamental elementary functions having been developed by Taylor's Theorem, § 32, we proceed to study some of the simplest operations that can be performed on series and thus, starting with the developments already obtained, pass to the developments de- sired. IV. ALGEBRAIC TRANSFORMATION'S OF SERIES. 34. It has been pointed out repeatedly (§§19, 21, 24) that since an infinite series is not a sum, ])nt a limit of a sum, processes appli- cable to a sum need not be applicable to a series ; if applicable, this fact requires proof. For exanii)le, the value of a sum is independent of the order in which the terms are added. Can this interchange in the order of the terms be extended to series? Let us see. Take the series 1 - i + ^- - i + Its value is less than 1 — A -)- a z terms as follows : 1 + i - i H- i + I - i + i -f The general formula for throe successive terms is («) -- t (§ 12) Rearrange its 1 I 111 CI (/?) 4A; — 3 ' 4fc— 1 2A; and if each pair of positive terms be enclosed in parentheses : (i + :U-.^ + (.^ + 4)-i + (^ + TV)-i+ (y) the result is an alternating series of tlie kind considered in (§11). For it is easy to verify the inequalities Hence the series (y) converges toward a value greater than (1 -|- |^) — A = f . The sum of the first n terms of (^) differs from a properly chosen sum of terms of (y) at most by the first term of a parenthesis, — a quantity that ai)proaches as its limit when » ^ x . lleuee the series (/3) and (y) have the same value and the rearrangement of terms in (a) luis thus led to a series (/3) having a different value from (a). In fact it is possible to rearrange the terms in (a) so that tlie new series will have an arbitrarily preassigned value, C. For, if C is positive, say 10 000, liegin by adding from the positive terms 1 + ^ + ^H- 44 ALGEBRAIC TRANSFORMATIONS OF SERIES. §§ 34, 35. till enough have been taken so that their sum will just exceed C. This will always be possible, since this series of positive terms diverges. Then begin with the negative terms 1 1 1 9 4 4 E" and add just enough to reduce the sum below C. As soon as this has been done, begin again with the positive terms and add just enough to bring the sum above C; and so on. The series thus obtained is the result of a rearrangement of the terms of (a) and its value is C. In the same way it can be shown generally that if ?/0 + «1 + «2 + is any convergent series that is not absolutely convergent, its terms can be so rearranged that the new series will converge toward the pre- assigned value C. Because of this fact such series are often called conditionally convergent, Theorem 1 of § 35 justifying the denoting of absolutely convergent series as uncoyiditionally convergent. There is nothing paradoxical in this fact, if a correct view of the nature of an infinite series is entertained. For a rearrangement of terms means a replacement of the original variable s^ by a new vari- able .s',^ , in general unequal to s^ , and there is no a x>Tiori reason why these two variables should approach the same limit. The above example illustrates the impossibility of extending a priori to infinite series processes applicable to sums. Most of such processes are however capable of such Q-s.ie.nB\o\\ under ^rroper restric- tions, and it is the object of this chapter to study such extension for some of the most fundamental processes. 35. Theorem 1. In an absolutely convergent series the terms can be rearranged at j:)leasure without altering the value of the series. First, suppose all the terms to be positive and let Sn = '^0 -{- ^h -\- + w„-i ; lim s„ = U. n = CO After the rearrangement let S'„' = < + u\ -\- + u\^,_i . Then s'„. approaches the limit U when n' = co . For s\^. always in- creases as n' increases ; but no matter how large n' be taken (and then hold fast), n can (subsequently) be taken so large that s^^ will include all the terms of s\^' and more too ; therefore § 35. ALGEBRAIC TRANSFORM ATIOXS OF SERIES. 45 or, no matter liuw hirgu a' be taken, S'n' < U' Hence .s'^^, approaches a limit U' 5 ' 2'' 9* 99 ' '2^^ 9* ' • • • Theorem 2. Jf U= ?/o + "i + V=v,-^v,-\- are any two convergent series, tlieij can he added term by term, or U^ V= vo + r, + », + ,., + ..,+ If they are absolutely convergent, the third series ivill also be abso- lutely convergent and hence its terms can be rearranged at pleasure. Let ^. = "o -h "i + 4- ",.-1. Then •*'„ + fn = («0 + ^-o) + ("1 -f ^'0 4- + (»„-! 4- i'„-l). 46 ALGEBRAIC TRANSFORMATIONS OF SERIES. § 35. When n =z CO , the left baud side converges toward U -{- V\ hence U+ V= («o + ^'o) + (^h + t'l) + It remains to show that the parentheses may be dropped. This is shown in the same way as in the case which arose in § 34. The proof of the second part of the theorem presents no difficulty and may be left to the student. Exercise. Show that if U = "o + wi + "2 + is any convergent series, c any number, CU =^ CUq -^ CUi -{- c n^ -\- Theorem S. If U = »„ + "1 -f ?'2 + y = ''o -r '"i + '"2 -r are any two absolutely convergent series, they can he multiplied together like sums; i. e. if each term in the Jirst series be multiplied into each term in the second and the series of these 2^>'oducts formed, this series will converge absolutely toward the limit UV. For example UV= UqVq -\- «o^'l + "I'V + "o'"2 + »l''l + ''2'-"0 + This theorem does not in general hold for series that are not absolutely convergent. Let .s„ = ?/o H- »i + + ^/„_i , K = ''0 + ''i + + ^„-i ; then liin .s t = UV. n li W = 00 The terms of the product s^t^^ are advantageously displayed in the following scheme. They are those terms contained in a square n terms on a side, cut out of the upper left hand corner of the scheme. -••■■' i .-•'' ' ••'■'^ I .-;■'' /""^ ,■-■■' .--' I . ' I ~ .•■" I -•■' ' The theorem asserts that if any series be formed by adding the terms of this scheme, each term appearing in this series once and § 35. ALGKHUAIC TKANSFOIiMATIONS OV SRKIES. 47 only once, — for example, the terms that lie on the oblique lines, the successive lines being followed fiom top to Ijottom : Wo^'o + "o^'l + "i^'u + "o''2 + , (a) this series will converge absolutely toward the limit UV. It is siitllcient to show that one series formed in the prescribed way from the terms of the scheme, for example the series formed by fol- lowing the successive boundaries of the squares from top to bottom and then from right to left, namely the series converges absolutely toward the limit UV. For any other series can then be generated by a rearrangement of the terms of this series. Let S_y denote the sum of the tirst N terms in (/3). First suppose all the terms of the /(-series and the v-series to be positive.* Then, if n^ < iVr< (n + 1)^, ^nK S ^■'' S ^'' + l^n + l- Hence lim 8^= UV. N= 00 Secondly, if the w-series and the v-series are any absolutely con- vergent series, form the series of absolute values «'o + «'. + "'. + v\ + v\ + c', -h The product of these series is the convergent series w>''o + "'o*"'i 4- "'it"'u + "'o«''2 + But this series is precisely the series of absolute values of (a), and therefore (a) converges absolutely. It remains to show that the value toward which it converges is UV. Since Sy approaches a limit when jV, increasing, passes through all integral values, S^ will continue to approach a limit, and tiiis will l)e the same limit, if ^^ passes only through the values }r : lim Sy = lim S„2 . JV = CO n = 00 But S„, = .-?,.^. and lim S„, = UV. n = x> This proves the theorem. ♦ The case that some of the terms are must not however be exchided ; hence the double sign ( ^ ) in the inequality below: Sy "^ s„ + \t„ + \. 48 ALGEBRAIC TRANSFORMATIONS OF SERIES. §§35,36. For example, let f{x) = tto + a^x -\- a^x"^ -f , ^(x) = &o + ^1-^' + ^2^^ + be two convergent power series, x any point lying at once within the region of convergence of both series. Then the product of these series is given by the formula This formula can be used to give the square, or by repeated appli- cation, any power of a power series. Thus it gives as the square of the geometric series 1 + •-« + ^^ + the series 1 4- 2 X- -f- 3 x'2 -(- . . . . . , a result agreeing with the binomial expansion of (1 -j- x)~^. Exercise. Find the first four terms in the expansion of X , log (\ -\- x') and »v -r y V 1 — .^■' 1 + ^ Square the series for e-^ and show that the result agrees ^ith the expansion of e^. 36. One more theorem is extremely useful in practice. Its proof would carry us beyond the bounds of this chapter. Let ^niy) = &0 + Wy + hy' + + Ky" be any polynomial in y and let y be given by the convergent power series in x : Then the powers of y : ?/'^, y^, y" can be obtained at once as power series in x by repeated multiplications of the .r-series by itself, the terms of the polynomial ^^ (y) then formed by multiplying these power series respectively by the cofficients 6, and the polynomial ^^ (y) thus represented as a power series in x by the addition of these terms. Suppose however that instead of the polynomial <}>^ (y) we had an infinite series : {y) = h-\- Ky + ^2y'' + Under what restrictions can the above process of representing (y). Suppose a^ = \ then this condition is always satisfied. And now our theorem is precisely this, that no further condition is necessary. Theorem 4. If «o = 0» '"-' f ether restriction is necessary; i.e. the above process of rejjresenting (y) as a jyoiver series in x is alicays applicable.* Remark. The point of the theorem just quoted is this. We know from § 35 that each term in the y series can be expressed as a power series in x : b„y" =fix) z= ao^") + a,^"^x + a^C'^x'' + and hence that (y) can be expressed in the form ^(^y)=f^(x)-\-f{X)+f,ix)-j- It remains to prove (and it is precisely this fact that the theorem as- serts, — a fact not true in general of a convergent series of the form fo{x)-{-A{x)+f{x)-\- , where f„(x) denotes a power series) that if we collect from these series all the terms of common degi-ee in x and then rearrange them in the form of a single power series, first, this series will converge, and secondl3% its value will l»e <^ (,'/)• Examples. 1. Let it be required to develop e*»'"^ according to powers of x.t Let y = x sin.r. Then cl^U,) = e" = 1 + ,/ + hir + i/ 4- iii/' 4- ,, ,.2 1 )•■! _1_ 1 1-6 1 r* -4- il/'= ^^' — tV^'+ ^y'= ^^'-^ ♦ The case n„ = is the oni' tliat usually arises in practice. But the theorem still holds provided only that — r <^ a„ ■< r, the only difference being that the coettifionts in the final scries will then be infinite series instead of sums. Cf. Stolz. Allgemeine Aritlimfiik. Vol. I, Cli. X, §25. t Even wlien it is known tliat a function can be developeii by Taylor's Theorem (v. Ch 111: Diff. Cal., Ch. IX; Ini. Cat.. Ch. XVin it is usually simpler to dt'ltrinini' the coefficients in the series by the method hero set forth than l)y performing the successive differentiations requisite in the application of Taylor's formula. The example iu liand illustrates the truth of this statement. 50 ALGEBRAIC TRANSFORMATIONS OF SERIES. § 36. 2. Find the first 4 tei-ms in the expansion of sin (Zc sin a;). 3 . Obtain a few terms in the development of each of the following functions according to powers of x. log cos a;. Suggestion. Let cos a- =z I -\- y- then y = — i -^"^ + 2T •'^■^ — and log cos x = ^ ^x^ — t^^* — 4T -*^^ "l~ 1 1 V 1 — 2.t; cos ^ -t- a;2 ^ i _ A:2 sin2a; Theorem 4 gives no exact information concerning the extent of the region of convergence of the final series. It merely asserts that there is such a region. This deficiency is supplied by an elementary theorem in the Theory of Functions.* But for many applications it is not necessary to know the exact region of convergence. For example, let it be required to determine the following limit. log COSX -j- 1 =^===: lim V 1 + ^^ + '^ x = sin.x — X Both numerator and denominator can be developed according to powers of x. The fraction then takes on the form ^ x^ -\- higher powers of x — ^x^ -\- higher powers of x Cancel x^ from numerator and denominator and then let x approach as its limit. The hmit of the fraction is then seen to be — 3. The usual method for dealing with the limit 0/0 is applicable here, but the method of series gives a briefer solution, as the student can readily verify. Example. Determine the limit lim s/ a^ — ^^ — V «^ + ^'^ x = 1 — cosa; An important application of Theorem 4 is to the proof of the following theorem. * Cf. Int. CaL, §220; Higher Mathematics, Ch.VI, Functions of a Complex Variable, by Thos. S. Fiske ; John Wiley & Sons. § 3(). ALGEBRAIC TKANSFOKMATIOXS OF SEHFES. 51 Theorem. The (jnotient of two power series can he represented as a poicer seiies, jn-oridcd the constant term in the denominator series is not : 60 + b^x. -\- boX- -f _ ^ j j o — i — (^0 ~\~ Cl^ ~\~ C2X -4- , fto + "i-c + Wo'*- + if fto -|= 0. It is sufliciciit to show that 1 can lie so represented, for tlien the power series that represents it can be multiplied into the numerator seiies Let y = a^x -\- a.^x"^ -{- 1 _1 1 _l_J/__i_l^_^_l_ provided y/uo is numerically less than unity, i. e. y numerically less than a^). Thus the conditions of Theorem 4 are fulfilled and the func- tion l/(ao -|- y) can be expressed as a power series in x by develop- ing each term ( — 1)" 2/"A^, + 1 i'^to such a series and collecting from these series the terms of like degree in x. CoROLLARV. //' the coefficients of the Jirst m poicers of x in the denominator series vanish, the quotient can be expressed in the form /... + i^ .,• + i, .r^ + _ C'_ (7-,„^x 4. ^ _U a 0(^4- a .10;"' + ^ -h iC" ' .T"'-i ' ' X For b^ -\- b^x -\- b^x^ -\- _ 1 61 -(- ^1 •^' + ^2'^'2 + 1 (Co-[-Cx.i-+o,.r+ ) X" and it only remains to set c^ =1 ^„-m ^"^ divide x"' into each term. Examples. Show that 1 2 tana; = x -t- ^ .i-^ + ^ .r^ + , ctn.r =z .1- — T^ .t + .r 3 4o and develop sec x and esc x to three terms. A more convenient mode of determining the coetlicients in these expansions will be given in § 38 . y. conti:n^uity, integration and DIFFERENTIATION OF SERIES. 37. Continuity. We have had numerous examples in the fore- going of continuous functions represented by power series. Is the converse true, namely, that every power series represents, within its interval of convergence, a continuous function? That this question is by no means trivial is shown by the fact that while the continuous functions of ordinary analysis can be represented (within certain limits) by trigonometric series, i. e. b}^ series of the form (/o -|- «i cos X -\- a^ cos 2 a; -|- -|- &iSina; -f- ^2 sin 2 cc -|- a trigonometric series does not necessarily, conversely, represent a continuous function throughout its interval of convergence. Let us first put into precise form what is meant by a continuous function. <^(a:) is said to be continuous at the point x^ if lim ^ {x) z=. (J3 (Xq) ; i. e. if, a belt being marked off bounded by the hues y = (I>(xq) -|- e and y =^ (x) will lie within this belt. These conditions can be expressed in the following form : § 37. CONTINUITY, INTEGRATION, Diri'KKENTIATION. 53 or * I <^ (x) — «^ (a-o) I < c , \ x — Xo\ < 8. A simple sulKcient condition that the series of continuous functions ?<„ (x) -\- 7ii{x)-\- represent :i contimious function is given by the following theorem. Theorkm 1. If "o (x) -h iiii^) -{- •. a < X < /8 , is a .series of cuntiniious functions couceryent throuyhout the interval (a, /?), then the function f (x) represented by this series will be con- tinuous throuyhout this inferral, if a set of positive numbers, Mq, J/^, M2, , independent of x, can be found such that 1) I a,^(x) I < 3/ , ao CIqCo 61 = OiCo + CloCi 60 = Oo-^o H- «'iCi + aoC2 A simple mode of solving these equations for the successive c's is furnished by the rule of elementary algebra for dividing one poly- nomial by another, Qiiotient: [ q^^ _|_ g^ x -\- C^X^ -\- &0 x-\-'bo «._, Co x''-{-h, x^-\- ^0 H" «i ^ H- «2 ^^ H" ^z ^^^ -\- (pQ — «o Co) + {bi — tti Co) «oCi ^+(^2 — 02^0) a'''+(&3 — «3fo) iC^-f- (&1 — aiCo — aoCi).T -|- (62 — 02'-'o — «iCi) ^'0^2 '^'' H- (&3 — «3('o — «2Ci) a;^ H- ttjC2 {h. — a.,r^ — a^(\ — a^,c^)x^-\-{b^ — a^c^ — a^c^ — a^c.^x^-\- etc. The equations determining the c's are precisely the condition that the first term in the remainder shall vanish each time. For example, to develop tan .t, divide the series for sin x by the series for cos x. Quotient: \^ ^ -\- ^ X^ -\- ^^ x"- -\- ^ "(T •*-" I T2'T7 '^ r i ^ *^ ~\ TTX **^ ^X 3 '^ I ^^- i x^^ ete. Hence tan X' ^ a; -(- ;^ .c^ ~h t^' ^^ ~f~ This method is applicable even to the case treated in the corol- lary, § 36. Exercise. Develop 1 \2 — bx-\- x^ 1 + a;' 3 + .r -(- a;' .7 ' csCa; . §39, CONTINUITY, INTEGRATION, DIFFERENTIATION. 57 39. The Integration of Series Term by Term. Let the eontinuou3 function /(x) be represented by an infinite series of continuous func- tions convergent througliout the interval (a, /?) : f{x) = n, {x) + n, (X) -h , a < X < ^ . {A) The problem is this : to determine when the integral of f{x) will be given by the series of the integrals of the terms on the right of equation (^1) ; i. e. to deterniinc wlioii f^f(x)dx= rn,{x)clx-{- l\i,{x)dx-\- {B) will be a true equation. The right hand member of {B) is called the term by term integral of the w-series. Let s,{x) = v^ix) + «iO«) -f + «„-i(a;), f{x) = s„(x)^r„(x). Then f^f{x)dx= f^s,Xx)dx^ rr„{x)d t/a t/a tJa or Jf(x)dx=z I UQ(x)dx-\- j rii(x)dx-\- • ' + / r,Xx)dx. Hence tlie necessary and sufficient condition that (B) is a true equation is that hm f^r(x)dx=0. To obtain a test for determining when this condition is satisfied, plot the curve y = r„ (x) . X + X'""-' (x)dx y OL » y- r.(X) V — Z'n Fig. 9. The area under this curve will represent geometrically i r„(-v)dx. 58 CONTINUITY, INTEGRATION, DIFFERENTIATION. § 39. Draw lines through the highest and lowest points of the curve parallel to the x'-axis. The distance p„ of the more remote of these lines from the .T-axis is the maximum value that | /•„ (x) | attains in the interval. Lay off a belt bounded by the lines y = p,, and y =: — p„. Then the curve lies wholly within this belt and the absolute value of the area under the curve cannot exceed the area of the rectangle bounded by the line ?/ = p„ , or (/? — a) p„ . This area will converge toward as its limit if * lim p„= 0, and thus we shall have a sufficient condition for the truth of equation (B) if we establish a sufficient condition that the maximum value p„ of I ?•„ (x) I in the interval (a, /3) approaches when n =i cc . Now we saw in the proof of Theorem 1 that if the series (A) satisfies the conditions of that theorem, Hence any such series can be integrated term by term and we have in this result a test sufficiently general for most of the cases that arise in ordinary practice. Let the test be formulated as follows. Theorem 3. Series (A) can alivays be integrated term by term, i. e. Jf(x)dx=: I UQ(x)dx-\- I Ui(x)dx-\- will be a true equation, if a set of positive numbers Mq, Mi, M^, , independent of x, can be found such that 1) I >^:X^) I < -^4, a(x) he a continuous function of x ayid let its maximum and minimum values lie between — r and r ivhen a '^ x \ fi. Let the power series converge lolien — r <^ y <^r. Then the series f{x)= ao-\- ai(x)-{- ao[cl>{x)Y^ can be integrated term by term from a to /3 ; Jf{x)dx = ao I dx-\-ai j (f>(x)dx-\-a2 I [cl>(x)Ydi For if Y be so taken that it is greater than the numerically greatest value of <^ (cc) in the interval a ^ a- ^ /3, but less than r, then 1) I «„| I <^(.i-) I » < |a„ I Y\ 2) I tto \+\ch\y+\ch\Y'+ converges ; and if we set I a„ I r« = M,, , the conditions of the test will be satisfied. Thus the integrations of §§ 24, 25 are justified. Lt + §§ 40, 41. CONTINUITy, IXTEGKATION, DII TEUENTIATIOX. ()1 Third Application. If the function (x) and the series «o + "uV + "2.'/^ + satisfy the same conditions as in the precedinrj theorem and if ^{/(x) is any contimious function ofx, then the series f(x) = a^il/(x) -\- a,xp(x) (x) + a.,ip(x) \_(x)Y -\- can he inter/rated term by term. The method of proof has been so fully illiistratcd in the two pre- ceding iippncations that the detailed coiistruetioii of the proof may be left as an exercise to the student. This theorem is needed in the deduction of Taylor's Theorem from Cauehy's Integral. Examples. 1. Compute TT /x^e'^'dx, I yj smx dx. 2. Show that l£cos(xsiu (^x) : 4> (x) — u'o (x) -\- u\ (x) + We wish to prove that ct.(x)=f'(x). By Theorem 1 the function <^ (a.*) represented by the w'-series is con- tinuous and by Theorem 3 the series can be integrated term by term : (J3(x)dx:= I « 'o (a;) d X- -f- / v'i(;x)dx -\- a. fj a *J a. = {"oO«)— "u(a)} + {»i(-^')— «i(«)) + = f(x)-f(a). Hence, differentiating, (x) =z /'(x), q. e. d. Exercise. Show that the series cos X cos 3 X cos 5 x can be differentiated term by term. By the aid of this general theorem we can at once prove the follow- ing theorem. Theorem. A poiver series can be differentiated term by term at any point within (but not necessarily at a point on the boundary of) its interval of convergence. Let the power series be convergent when \ x \ <^ r, and form the series of the derivatives : «! -|- -la^x -f- 303.1-2 -f- Then we want to prove that if | Xq | <^ ?•, /'(a-o) = «! -\- 2a2.ro -h 3a3.V -f It will be sufficient to show that the series of the derivatives con- verges when I .X' I <^ ?• ; for in that case, if X be so chosen that I iCo I X and x" > X. We proceed now to the proof. Let us choose for the successive values that c is to take on any set Cj, cg*, £3, steadily de- creasing and approaching the limit ; — for example the values 1, ^, ^, ) «, = 1/i- Denote the corresponding values of X by Xi, X2, X3 Then in general these latter values will * For the notation cf. foot-note, p. 53. APPENDIX. 65 steadily increase, and we can in any case choose tliem so that they do always increase. Begin by putting e = q : I f{x') -/(.V) |< c, , .'.' > X\ , X" > X,. Assign to a;' the value X^. Then \f(X,)-fix) | ^-t. I 4—1 = 1*1!. : ', =:Pi A Fio. lu. Denote the left hand boundary f{Xi) — e^ of this interval by a^, the right hand l)Oundary f(Xi) -|- c^ by /Sj. Then, to restate con- cisely the foregoing results, «! < f(.^) < /3i if ^ > ^i ; A — ai = 2 €,. Now repeat this step, choosing lor c the value co ' |/(X,)— /(.1-) | X2 ; /?2 — a2 < 2 e^ . The remaiudev of the proof is extremely simple. The step just described at length can be repeated again and again, and we shall have as the result in the general case the following : Now consider the set of points that represent a^, ug, . . . a,, . . . They advance in general toward the right as i increases, — they never recede toward the left, — but no one of them ever advances so far to the right as /Jj. Hence, by the principle* of § 4, they approach a limit A. Similar reasoning shows that the points repre- senting /3i, /^a, . . • A, • • • approach a limit B. And since these limits must be equal : A = B. From this it follows that f{x) converges toward the same limit. For and if when x increases indefinitely, we allow i to increase indefi- nitely at the same time, but not 1-0 rapidly as to invalidate these in- equalities, we see that /(a;) is shut in between two variables, a, and ;8,-, each of which approaches the same limit. Hence /(a;) approaches that limit also, and the theorem is proved. In the theorem in infinite series above quoted n is the independent variable x, s^ the function f(x) ; the expression s^^^ — s^ corre- sponds to f{x') — /(a;"); and thus that theorem is seen to be a special case of the theorem just proved. The domain of values for the variable x is in this case the positive integers, 1, 2, 3, Another application of the present theorem is to the convergence of a definite integral when the upper limit becomes infinite. Let f(x)= rct>{x)dx. tJ a 4> (x) d X— I (x) d X. J'' a-/ (^ {x) dx =z * This principle was stated, to be sure, in the form 5',/ I> 5 if ?;'> w; but it obviously continues to hold if we assume merely that S,,' ^ S,, when n' > n. APPENDIX. 67 when x' and a;", regarded as independent variables, both become in- finite, tlie integral f (ji (x) dx is convergent. The domain of valnes for the variable x is in this case all the real quantities greater than a. In the foregoing theorem it has been assumed that the independent variable x increases without limit. Tiie theorem can however be readily extended to the case that x decreases algebraically indefi- nitely or approaches a limit a from either side or from both sides. In the tirst case, let x = —y; in the second, let , 1 X =z a -\ — y if X is always greater than its limit a ; let 1 X =^ a y if X is always less than a. Then if we set f(x)= c/>(.v) and the function ^ (//) satisfies the conditions of the theorem when y =z -\- cc , (y) and hence /(.r) will approach a limit. Finally, if x in approaching a assumes values sometimes greater than a and some- times less, we may restrict x first to approaching a from above, secondly from l)elow. In each of these cases it has just been seen that /(a;) approaches a limit, and since lim[/(.i•')-/(.^•")] z=0 where x' and x" may now be taken the one above, the other below a, these two limits must be equal. We are thus led to the following more general form of statement of the theorem. Theorem. Let f(x) he such a function ofx that Urn [/(x')— /(x")] = when x' and x", regarded as independent variables, approach the limit a from above or from below or from both sides, or become jwsitively or negatively infinite. Then f(x) apjrroaches a limit when x approaches the limit a from above or from below or from both sides, or becomes positively or negatively infinite. 68 APPENDIX. A TABLE OF THE MORE IMPORTANT FORMULAS. The heavy line indicates the region of convergence. ._J_ = 1 + a; + x-^ + ..3 + — 1 1 a — bx a cr cr cr ■r II r r = - numerically. x^ , x^ X* log (1+X)=.T -- + --- + — 1 1 (^ + f + f+ ) log [^=2(0. + :^+:^ + — 10 1 (l + x)M=l+^. + tl^..-^ + ?iOi=i|i^.^ + — 1 ^ = 1 _ 2ic + 3.^•2 — ix^ -\- {l-\-x) — 1 APPENDIX. 69 I O ' ■> . 1 I •> . /I . c I V 1 — a;2 '2 '21 ' 2 • 4 • 6 — 1 1 Vl_a-»^l-^x»-^j^-^.. — 1 (I 1 /V.2 ,,.8 ,.4 «^= 1+^ + ^ + 3. + .!,+ a;* , x^ x'' sinx = x---^^--^ II a;2 , a;* .j;« cosa.'= i__-f + 4! 6! tan a; m a; H- ^ .r^ -f- j-^ x^ -|- - ^ •■> cot ar — ^ .)• — \ x^ -\- X ■ IT < ' E 70 APPENDIX. sec x=l-\-^x^-\-^x*-\- ' • — TT Y . , , 1 a;3 1 . 3 a;5 , 1 • 3 • 5 a;'' , — 1 1 x^ , x^ .v'' tan ^x r= X 1 '■ — 1- 3 i — 1 1 f(Xo-\-?l) = /(Xo)-\-f'(x,)h-\-r(Xo)^^ H- +/(«)(a:„ + ^/,) ^" 0<^<1 f{x, -j-h)= /(.i-o) + hf (a-o 4- ^/i) Jo ^/ 1 — k'^sin^ TT E = j ^ I — k'^sm^ff, d = Ari'ENDIX. 71 For small values of x the following equations are approximately correct. f(a + x)=f{a)-\-f'ia)x (1 -^ x)"'= 1 -(- ma; (I -\- xy = \ -\- -Ix VT + ^rrr 1 + ^x = \ — 2x ii-\-xy , ^ = 1 — Ax V 1 + cc If X, y, z, to, are all numerically sn.all, then, approximately, (1 + .T) (1 + 2/) (1 + ^) = 1 4- .0 + y + ^ + sin a; == x or x — ^ x^ cos X rr: 1 Or 1 — i .i'^ tan X' z= X- or .r -f- j^ x^ sin (a -|- a*) — sin a = x cos a cos (a -|- x) — cos a = — x sin a log (a -\- x) — log a r= - ! nr. '^nuTHfRfi RFGIONAL LIBRARY FACILITY nil AA 000 507 745 b