TAYLOR'S THEOREM. 



TAYLOR'S THEOREM. 



In the Penny Cyclopaedia' we gave a comparison of five proofs of 

 Taylor's Theorem. This was twenty years ago : since which time 

 the character of elementary works has changed. The Penny Cyclo- 

 paedia being still perfectly accessible, we think it will be best to confine 

 ourselves, in the present work, to a statement of the best form of the 

 best proof. This we hold to be a variation and amendment, by Mr. 

 Homersham Cox, of Cauchy's proof, which may be seen in De Morgan's 

 Differential Calculus. Mr. Cox's proof was first published in the 

 Cambridge Mathematical Journal, vol. vi., p. 80. 



From <p(a + 1>) subtract any number of Taylor's terms, and one more 

 with an undefined constant, and write this down with as many dif- 

 ferential coefficients as Taylor's terms : as in 



' t? v* 



' 



2.3.4 



Q <t>'(a + r) - <t>'a <t>"a v <t>'"a -g - c g-jj 



/ : 

 B <t>"(a + r) <t>"a <t>'"a v - c y 



s <t>'"(a + v) <t>'"a ce 

 T $>'(< + r) c. 



All these vanish with v, except the last : choose for c that value 

 which makes the first vanish when e=A. Let <f>x be such that <f>" s 

 (and consequently $x, ifi'x, <t>"x, <p"x) does not become infinite from 

 x=a to x=a + A. Now remember that a function which vanishes in 

 two places, and does not become infinite in the interval, must change 

 from increasing to diminishing, or from diminishing to increasing, in 

 that interval : so that its differential coefficient must change sign, and, 

 if not infinite, must vanish. Now P satisfies these conditions, vanish- 

 ing at r = and at r = A : hence q must vanish before r= A, and as it 

 vanishes at = 0, and does not become infinite in the interval, Q also 

 satisfies the conditions : hence R vanishes before r=A : and by like 

 reasoning s, and T. Now if some value of v between and h makes T 

 vanish, let it be v = 8h, B being a positive fraction between and 1. 

 Hence c=<f>"(a + eh): and since c was BO taken that p vanishes when 

 =/i, we have 



X s 7i 3 h 4 



f (a + A) = <t>a + fa . h + <f>"a ^ + <t>'"a g-jj + <p*(a -r flA) j jj J. 



This, if the reasoning be carried to n terms of Taylor's series, is 

 Taylor's theorem with Lagrange's theorem on the remainder of the 

 series appended. If no differential coefficient of <fx up to <t>(">x should 

 become infinite from x=ata x=a + k, then 



*' 



If the remainder term diminish without limit as n increases without 

 limit, Taylor's series gives a true development. 



Sumo views of Lambert on the reduction of the roots of equations 

 (Acta Helvetica, 1758) into series were generalised by Lagrange 

 (Mem. Acad. Sci., 1768) into a celebrated theorem of development 

 bearing his name ; and this again was generalised in form by Laplace 

 (M?c. C(S1.). The problem is as follows : given 



y=T(s + x<l>y) ---- (A) 



required the expansion of t^y, when possible, in powers of x. Since 

 l>y the preceding equation, a function of x and :, if z be constant, 

 and we differentiate with respect to x, and then make jc = 0, or i = tz, 

 we may use Stirling's theorem. But this differentiation would be 

 laborious and indirect ; it was made more direct (by Laplace) in the 

 following manner : A constant may have any value given to it, or may 

 be made to vanish, either before or after differentiation with respect to 

 a variable : if then we can express differentiations with respect to x in 

 terms of differentiations with respect to z only (in which x is constant), 

 it will be in our power to make x vanish hefure the differentiations, 

 which will reduce the indirect or implicit to direct differentiation. 

 This substitution of c-differentiations in place of those of x is done as 

 follows : Differentiate (A) both with respect to x and i separately, and 

 we have 



^ = F- (; + x<t>y) { ^y + xfy -I J. whence 



g -- 



Let u be a function of y only, that is, not of a: or z except as those 

 variables are contained in y : then 



du dy da, dy du da, 



-' = 



Fn<rn this equation only it may be shown (by INDUCTION) that 



rf' 



<**-> 



du 



as follows. Assume the preceding to be tiue for one value of n, and, 



since ($>/) * x du : dy is a function of y only, let it be dv : dy, v being 

 another function of y. 



d*u d'~ l /dv dy\ d* v 

 dx' = d2*=i Ufy A J = rfF 



Z* rdv 

 = 



dv 



du dy 



dv dy 

 Ty &> 



du 



whence the theorem remains true after writing n + 1 for n. But it is 

 true when = 1 ; therefore it is true for all values of n. If then we 

 make x 0, or y = Tz, which may be done before the differentiations on 

 the second side of the equation, We have (u being 



d 1 



Apply this to Maclaurm's Theorem, and we have Laplace's Theorem, 

 namely, 



y = F (z + xtfry) gives if)/ = 

 d 



difiFZ\ .r" 

 - 



the general term, ^-f 



Lagrange's theorem, from which Laplace generalised, is the case in 

 which fx=x; namely, 



ves if,y = <f>z + ($z$'z)x + -^ ( (f- r ) 2 f 



tZ"-' r i x* 



the general term ^n | (<>z)fz |-^ 



^ + , &c. 



y = z + QZ. 



dz- 2 . 3 



+,&c. 



Lagrange's Theorem leads to Hermanns Theorem (presented to the 

 institute in 1796). The second is in fact the same as the first, though 

 very different in form, and arrived at independently. It is required, 

 when possible, to expand $x in powers of $u\ This might be done 

 indirectly, by expanding i^" 1 x in powers of x, and substituting <t>x for 

 x in the result. The form in which Burmann obtained Lagrange's 

 theorem avoids the indirect process. Let ^x vanish when x= a, and 

 let <fx (x a) : x x i or z=a + <px . \x. We can now employ La- 

 grange's theorem to expand ifac in powers of <t>x, and we have 



d I \ (<t>x)* 



ft = ifa + X af a . <t>x + -^ ^(x)" <K a ) %-+,&. 



Now the general term of this has for its coefficient the value of 



d ' 



whena:=o : consequently i|u", expanded in powers of Qx, is found by 

 making x = a in the coefficients of the powers of <px in the folloiving 

 series : 





d 



+ , &c. 



When, in a function of any number of variables #,, x,, &c., the varia- 

 bles are severally to receive increments A,, A 2 , &c., the law of the 

 development is best seen by the calculus of operations. [OPERATION.] 

 To change x into .r + A is to perform the operation t*", D being the 

 symbol of differentiation with respect to x : the condensed form of the 

 development now before us is 



,, ..... 



where D,, D.,, &c. refer to x lt x,, &c. The general term of the develop- 

 ment is 



(A,D, +AjD, + ..... )" 



- 



which must itself be developed. It is not worth while to pursue this 

 case further : we shall only observe that when it is desired to stop, 

 the remnant may be obtained by writing in the last term x l + 0A, for 

 ar,, a-, + flA, for x v &c., where fl, the same in all, is either or 1 or 

 between them. 



The value of x which makes $x = Q is represented by 



__ 

 >' 2^" 



2.8.4 <?>'' 



2.3.4. 5<t>'' 



where a is any assumed value (the nearer the root the better) and 

 <t>, <t>, &c. represent <f>a, <f>'a, &e. This scries is obtained by common 

 reversion from <p(a + h) =0. For the forms which Paoli gave to this 

 series, and also to Burmann's, see Lacroix, vol. i., pp. 306-308. The 



