TAYLOR'S THEOREM. We propose in this part of the article to give some results of the methods of algebraical development which are consequences of the celebrated theorem, the history of which is given in the article TAYLOR, Benoit, in Moe. Div. The simplest parts of the Differential and Integral Calculus will be presumed known. It is not usual in works on that subject to bring 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 done in a work of reference. It is to be particularly remembered that we do not here profess to teach the subject of development, but only to recall the steps of the several processes to those who have already learnt them, and to present the theorems in a form which can be easily referred to.
As to notation, we shall frequently signify differentiation by accents : thus ex is the second differential coefficient of cox with respect to x ; is the third differential coefficient of the product of ox and 4,.r. And [a] will signify the product 1 x2x3x x (n-1) x u. Moreover, when a series is written, three terms will be written down, and the general tens appended.
Taylor's theorem is as follows : /1.
+ A) = iox co'x sb".r + (p(*).c This theorem is true whenever x has such a value that-I. No one of the set ˘x, 6:,'x, &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=d In the first function, o's, and all which follow, are infinite when x= a ; in the second cpx and all its differential coefficients vanish when x=n. The meaning of this circumstance is as follows : the form of Taylor's theorem essentially requires that 4.(x+ A) should be developed in ascending integer powers of A; consequently when such form of development Is impossible, this theorem may show signs of being inapplicable. Now, the first of these functions (when x=n) can only have ˘(a+ h) expanded in ascending fractional powers; and the second only in descending integer powers. Those who will only allow the
time of converging series may require also that h should be so small that the resulting series is convergent : but this Cauchy has proved always happens if A be small enough.
In the Penny Cyclopmdia' 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 pmdia 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'a proof, which may be seen in De Morgan's Differential Calcnlus. Mr. Cox's proof was first published in the Cambridge Mathematical Journal, vol. vi., p. SO.
From cp(a + r) 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 All these vanish with v, except the last : choose for C that value which makes the first vanish when v =h. Let ox be such that x (and consequently ox, V"x) does not become infinite from to + h. Now remember that a function which vanishes in two 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 r satisfies these conditions, vanish ing at r=0 and at v=h : hence Q must vaoisb before v= h, and as it vanishes at v=0, and does not become infinite in the interval, Q also satisfies the conditions : hence R vanishes before r : and by like reasoning a, and T. Now if some value of v between 0 and IL makes T vanish, let it be v.-- Oh, 0 being a positive fraction between 0 and 1. Hence c=ssi' (a+ Oh): and since o was so taken that P vanishes when we have 4 cp(a + 1)= (pa + • h + cp"a + fp"'a + (a + Oh) h.