curvature and of motion, lies in immediate contact with its most interesting applications. The (4) When the remainder R, keeps indefinitely diminishing without limit as r increases, Taylor's series is convergent; the converse proposition is not, however, generally true. theorem therefore, regarded as an indefinite expansion, fails to express the true value of f(x + h), not only whenever the series is divergent, but also, which constitutes a still greater limitation, whenever R, does not diminish indefinitely with the increase of r. Stirling's Theorem. 90. If in the formula of Taylor we put x = 0, we get This theorem enables us to expand any function of x in a series ascending by positive integral powers of x. Its demonstration was given by Stirling in a work entitled Methodus Differentialis, London, 1730, p. 102. The theorem was afterwards given by Maclaurin in his Treatise of Fluxions, Edinburgh, 1742, p. 610; and is now generally called Maclaurin's theorem. It is in fact, as we see from the demonstration given above, merely a particular case of Taylor's theorem. Stirling's series, as well as Taylor's, from which it has been deduced, ought to be completed by a remainder. Let R, represent the remainder, when we stop at the (r + 1)th term of the development: then the value of R,' being given in the form of a definite integral by the equation a formula derived from the expression for R in Art. (84), by first putting ≈ = 0, and then replacing h by x in the result. Again, the limits between which R, always lies, are P', Q′, denoting respectively the least and the greatest values of fr1 (x − z), or, which amounts to the same thing, of fr1(2), between the limits z = 0, z = x. If 0, 0, represent certain unknown numbers comprised between 0 and 1, we shall have also for R,' the two expressions Examples of the Application of Stirling's Theorem. 91. Ex. 1. Suppose that f(x) = a*. Then f'(x) = log a. a*, ƒ"(x) = (log a)2. a*, ƒ""(x) = (log a)3. aa‚. . . . ƒ'(0) = log a, ƒ"(0) = (log a)2, ƒ'''(0) = (log a)3,. . . . . . We have then, by Stirling's theorem, will evidently approach without limit to zero, when r becomes indefinitely great, whatever be the value of x. Hence we are at liberty to put the series being supposed to be carried on to infinity. ƒ'(x) = (x + 1)`', ƒ'(x) = − (x + 1)2, ƒ'(x) = (-)2. 1.2 (x + 1) ̃3,... and generally ƒ′(x) = (-)TM*1. 1.2.3. .. .(r − 1) . (x + 1)*; whence ƒ(0) = 0, f'(0) = 1, ƒ"(0) = − 1, ƒ"(0) = (-)2. 1.2,. ..., ƒ'(0) = (-)"11. 1.2.3. .. .(† – 1). If x be positive, then the expression R', will diminish without limit with the increase of r, provided that the series being prolonged to infinity. If x be negative, let z represent its value: then and, in order that we may be sure of the propriety of regarding the series for log (x + 1) as infinite, we must have hence we are obliged to suppose that z is <, or that the negative values of x are comprised between – 1⁄2 and 0. If however we take the other formula for R',, we see that R', = (−)”+2 (1 − 0 ̧) ̃. (1 + 0 ̧x)*-1 xr+1 will be less than unity, provided that z − 0 ̧z < 1 - 0 ̧z, or z < 1. Thus we see that the latter formula for R', allows a greater range of negative values for x than we could have inferred from the former. Ꮖ If then x have any value comprised between the limits - 1 and + 1, we know that whence ƒ(0) = 0, ƒ'(0) = 1, ƒ"(0) = 0, ƒ"(0) = − 1, ... From these expressions it is plain that, when r = ∞, whatever be the value of x. Hence, generally, whatever be the value of x, sin x = + 1 1.2.3 1.2.3.4.5 the series being regarded as infinite. We might prove in the same way that, whatever x be, Then f'(x) = m(a + x)m-1, ƒ"(x) = m(m − 1) (a + x)m-2, n-1 whence f'(0) = a", ƒ'(0) = m aTM-1, ƒ"(0) = m(m fr (0) = m(m − 1) (m − 2)... (m − r + 1). am-r. |
