accordingly as r is odd or even. (-) cos (0x), From these expressions it is plain that, when r = ∞, whatever be the value of x. Hence, generally, whatever be the value of x, the series being regarded as infinite. We might prove in the same way that, whatever ≈ be, Then f'(x) = m(a + x)m-1, ƒ"(x) = m(m − 1) (a + x)m-2, ƒ''(x) = m(m − 1) (m − 2) (a + x)m-3, f'(x) = m(m1) (m − 2).... (m − r + 1). (a + x)TM+: whence ƒ'(0) = a”, ƒ'(0) = m am-1, ƒ"(0) = m (m − 1) am-2, = f'(0) = m(m1) (m2)... (m − r + 1). aTM-*. 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, 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 z(10) great, if < 1, ≈ (1 − 0 ̧) < a − 0 ̧ã, z<a. Hence, for all values of x comprised between - a and + a, m (a + x)m = am + am-1 x + 1 m (m − 1) am-2x2 + m(m-1)(m-2) am-3 x3 1.2.3 ad infinitum. Ex. 5. Let 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 f(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 that & x2 = 0 for all values of x. This shews that the mere convergency 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. 1 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', kak', 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). d p'(a): = dx )=h' 22 1. f(x + ah', y + ak') . h' + f(x+uh', y + ak'). k', or, to adopt a more concise notation, differentiating again, d dy p"(a) = h'2 for a third differentiation and so on indefinitely, the law of derivation being obviously in accordance with the binomial theorem: we thus have, generally, dx dy +k's d3f 39 From the expressions for (a) and 4"(a) it is plain that $(0)=ƒ, where ƒ is used to represent f(x, y), and p"(0) = h'" = h'n d'f n + h'n-1k' dxn 1 Hence, from (1), substituting for ø(0), p′(0), p′′(0), their d3f ...(2), L 2 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 hand k are any quantities whatever, we may put h = dx, f(x + dx, y + dy) = ε' or, since d + d = D, D denoting total differentiation, 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 |