Suppose that x = a positive quantity: then, in order that R,' may become zero, when r is made indefinitely great, since, as may easily be seen, m (m-1) (m2)....(mr) 1.2.3... (r + 1) If x be negative, it will be convenient to have recourse to the second formula of Art. (90) for R,, which gives Hence R,' will be reduced to zero, when r becomes indefinitely great, if ≈ (1 − 01) < 1, z(1 - 0) < a − 0 ̧z, z < a. a - 01z Hence, for all values of x comprised between - a and + a, and suppose that ¥(x) is a function the development of which, by Stirling's theorem, may be continued indefinitely. L the derivatives of being all zero when x = dent when we consider that 0, as will be evi is equal to zero for all positive values of m and n, a being any positive number greater than 1. Thus we see that in this example Stirling's series for the development of ƒ(x), when continued indefinitely, is convergent, and yet that it does not give for f(x) a true value. In fact it makes the development of f(x) the same as that of (x), from which it would follow 1 that & x2 0 for all values of x. = This shews that the mere con vergency of the series, although necessary, is not sufficient for its truth, there being an additional condition, viz. the convergency of the remainder R,' to zero. Extension of Taylor's Theorem to Functions of two Variables. 92. Suppose that in the function f(x, y), x and y are replaced by x + h and y + k; our object is to obtain a development of the function f(x + h, y + k) by ascending powers of the increments h and k. Putting hah', k = ak', we have f(x + h, y + k) = f(x + ah', y + ak'), which is a certain function of a, which we will denote by (a). dxdy dy3 and so on indefinitely, the law of derivation being obviously in accordance with the binomial theorem: we thus have, generally, + 3h'h2 d3f + 43 39 From the expressions for (a) and 4"(a) it is plain that 4(0)=ƒ, where ƒ is used to represent f(x, y), and Hence, from (1), substituting for (0), p'(0), p′′(0), ..... their values, and replacing ah', ak', by h, k, respectively, we have The formula (2) may be expressed more briefly by the aid of the separation of the symbols of differentiation from those of the function upon which they operate. Thus the propriety of these symbolical expressions depending upon the principles of Art. (48), which shew that the laws of the Since h and k are any quantities whatever, we may put hdx, or, since d2+ d f(x + dx, y + dy) = εo.ƒ COR. The method of development which we have applied to a function of two variables may obviously be extended to a function of any number of variables whatever. Thus Failure of the Development of f(x + h, y + k) by Taylor's Theorem. 93. The development of ƒ (x + h, y + k), given in the preceding article, fails for particular values of x and y, whenever f or any of its partial differential coefficients becomes infinite; this failure being consequent upon the failure of Stirling's theorem applied to the expansion of (a). It will likewise cease to be applicable for particular values of x and y, which render for its partial differential coefficients essentially indeterminate. Limits and Remainders of the Development of f(x + h, y + k). 94. Let R be the value of the remainder which must complete the series (1) of Art. 92, supposing this series to be stopped at the end of the (r + 1)th term; then, by Art. (90), 0, 0,, denoting certain unknown numbers comprised between 0 and 1. Hence, if we stop at the term of the series for the development of f(x+h, y + k), we must add the complementary remainder where Pr The numerical fractions 0, 01, which enter into these formulæ, being unknown, we cannot employ the formulæ for the actual computation of p, they serve only in fact to determine limits between which the value of p, must lie. |