T A Y 



127 



T A Y 



conn;rxron with it. For the use which Lagrange proposei 

 la make of it, see DIFFERENTIAL CALCULUS, and FUNCTIONS 

 THEORY OF. From the time of the publication of the work 

 cited in the article last referred to, Taylor's theorem take 

 that place which, if it had always occupied, we should no 

 have had to write any history of it. Full justice is don 

 to the discoverer : it only remains to restore to Stirling tin 

 view of the theorem which has hitherto been given to 

 Maclaurin, 



TAYLOR'S THEOREM. We propose in this part of fhe 

 article to giv e some account of the methods of algebraica 

 development which arc consequences of the cclebratec 

 theorem, the history of which is given in the last article 

 The simplest parts of the Differential and Integral Calculus 

 will lie presumed known. It is not usual in works on that 

 subject to brinsr together in one place the most conspicuous 

 theorems which have arisen out of that of Taylor ; which 

 makes it the more desirable that such a thing should be 

 in a work of reference. It is to be particularly re- 

 membered that we do not here profess to teach the subject 

 of development, but only to recall the steps of the M 

 processes to those who have already learnt them, and to 

 present the theorems, in a form which can be easily re- 

 ferred to. 



A-, to notation, we shall frequently signify differentia- 

 lion by accents : thus <Ji".i- is the second differential co- 

 efficient of 0r with respect to ./ : (<.ri|/.r)'" is the third 

 differential coefficient of the product of (fix and ij/.r. And 



[M] will signify the product 1X2X3X X it 1 X//. 



over when a series is written, three terms will be 

 written down, and fhe general term appended. 

 Taylor's theorem is as follows: 



' -=- + &c. 



This theorem is true whenever .r has such a value that 

 1. No one of the set </>i; tjj'.r, &c. is infinite. 2. All of 

 them do not vanish. Thus neither of the following could 

 be allowed to be treated by it when x=a : 



and 



~ ( 



In the first function, <'./, and all which follow, are in- 

 finite when .r = n : in the second tji r and all its differential 

 coefficients vanish when ./ -n. The meaning of this cir- 

 cumstance is as follows : the form of Taylor's theorem 

 :Mally requires that <j>(x-\-h) should be developed in 

 UCendinK integer powers of /' ; consequently when such 

 form of development is impossible, this theorem must show 

 - of being inapplicable. Now, the first of these func- 

 tions (when x=a) can only have $(a+A) expanded in 

 ascending fractional powers : anil the second only in de- 

 scending integer powers. Those who will only allow the 

 use of converging series may require also that h should 

 be so small that the resulting series is convergent: but 

 this objection will afterwards be inapplicable, as will be 

 seen. 



We shall state five proofs of this theorem briefly, being 

 substantially those given by Taylor, Maclaurin, D'Alem- 

 bert, Lagrange, and Ampere. 



Tui/lni'x I'rtuif. Let y = A,and form differences of <fi>- 

 from the series <j: <f,:.r+0\ <(.r+20), .... tf>(jc+>W). 

 juently we have [DirnBXNCZJ 



where Ax = 6. 



tfi.i- + H A<r -f- >i 



Throw this into the form 

 , h 



Arf>.r 

 - A+ 



, &c. 



, &c. 



without, limit, A.c at the same time dimi- 

 nishing, so that )<A.r remains always =A. Then 



Limit of 



-, &C. 



So that Taylor's theorem is proved when we know that 



<(> rtittl the lin>it of A 0.r : TA./-)'. This 



WM : iition* of Taylor's: but in the modern dif- 



' Si pro incrcoj' - '.TiUantur lluxioDet iji |iropor- 



'kiualn, Sic.' ?1h lUtrncnl In TATI.OH, Bloom 



ferential calculus it is a hetter plan to prove Taylor's 

 theorem in another way, and then from the preceding fol- 

 lows the simplest manner of showing the identity of 



< & and the limit of A <f>x : (A.r) . 



Maclaur iu's Proof. The method here given was first used 

 by Maclaurin, and though it was only applied to develop 

 <f>\ + h), yet it will do equally well for <j>(x+h) ; and Mac- 

 laurin himself saw no difference (as indeed there is none, 

 < being any function whatever) between the two cases. 

 It turns upon <t>(x+h) giving the same result, whether 

 differentiated with respect to .r or h, and assumes the form 

 of the development, which is a radical defect. It is as fol- 

 lows: Let^(.r + /()=A+BA-fCA !i -l-,&c.; then tf>'(x+h) = 

 B + 2CA + 3Vh* + , &c., 0"(;r + A) = 20 + 3.2DA +, &c. 



, ., . , . 



'" (.r+A) = 3.2D -j-,&c., which, when A = 0, give d>x= 

 , 4>'x = B, <f>"x = 2C, <t>"ijc = 3.2D, &c. ; from which 



the theorem readily follows. The common proof, given 



in most elementary works on the differential calculus, is 



but a less commodious form of this. 

 D'Alembvrt's Proof . The first principles of the Integral 



Calculus give 



n-|A />0 



<j>'xdx=- I 



(I */ /I 



he last step being made by parts. Similarly 



/* a it h * r k 



j 2 v o 



z'dz 



and so on : whence it appears that if we go up to A in the 



series, the term involving A may be followed by ano- 

 her, expressed in the form of a definite integral, and 



which alone represents all the remnant of the series ; as 

 bllows : 



*%4]^ /X + '' rt+/ '-^ v ~~- 



The conditions of integration require that neither 



r, <j>'.r, .... a? should be infinite from x = a to x-=a 

 +A, both inclusive : this one condition being satisfied, the 

 difficulty of divergent series disappears ; for the theorem 

 loes not give an infinite series at all, but only any number 

 ve please of the terms of a series together with a con- 

 luding quantity which is finite both in form and reality. 

 Phis integral might frequently be difficult to use, but limits 

 or its value may be readily obtained. Let P and p be the 



greatest and least values of <j> x from . = a to x = a 

 + /', both inclusive : then the concluding integral lies 

 jetween 



P/^z'dz fmdpf z'dz or p; '" andX . 



c/O v H-4-\ N -f- 1 



Now when a continuous function does not become in- 

 nite between two values of .r, every quantity which lies 

 jetween its greatest and least value is one of its interme- 

 iate values: or anything between P and p is a value 



(n +0A). for some value of which is either 



f 



" " ' } 



>r 1, or between them. Hence the preceding expression 

 nay be written 



The following form has been given by M. Cauchy. Let 

 and p represent the greatest and least values of <f> 

 + h - z} . z" from z=0to z =A, both inclusive: pro 

 .sely similar reasoning will give for the last term chosen 

 f Taylor's series, and the value of the remnant, 

 n ,"+i 



.00 h 

 g> a Y~ 



here e is either or 1, or between them. 



