Theorem V. A series is convergent when its terms are alternately positive and negative, provided each term is less than the preceding, and that the terms decrease without limit in absolute magnitude. Let the series be Uş +... ± Un ‡ Un+1 ± U By writing the series in the forms n+2 - F we see that, since each term is less than the preceding, the sum of the series must be intermediate to u, — u, and u1; and hence the sum of the series is finite. It is also similarly clear that the absolute value of U- U, is intermediate to the absolute values of un и and un and therefore U-U1 becomes indefinitely small when n is increased without limit. The series must therefore be convergent. n+1 n+2 n+1' the terms are alternately positive and negative and decrease without limit. 2 3 4 5 series although its sum is a finite quantity between and 2, for the The series is not however a convergent 1 2 274. We will now apply the preceding tests of convergency to three series of very great importance. I. The Binomial Series. In the binomial series, namely the number of terms is finite when m is a positive integer; but when m is not a positive integer no one of the factors m, m-1, m-2, &c. can be zero, and hence the series must be endless. To determine the convergency of the series when m is not a positive integer we must consider the ratio Hence, for all values of n greater than m+1, n +1, and u have different signs when x is positive, and have the same sign when is negative. Moreover, as n is increased, the absolute value of un+1/un becomes more and more nearly equal to x. If therefore be numerically less than unity, the ratio u+1/un will, either from the beginning, or after a finite number of terms, be numerically less than unity. Hence by Art. 267 the series formed by adding the absolute values of the successive terms will be convergent, and therefore also the series itself must be convergent, whether its terms have all the same sign or are alternately positive and negative. Thus the binomial series is convergent, if x is numerically less than unity *. II. The Exponential Series. In the exponential series, namely the ratio +1/ is x/n. Hence the ratio u+1/u, is numerically less than unity for all terms after the first for which n is numerically greater than x. therefore convergent for all values of x. The series is * The series is also convergent when x=1, provided n> convergent when x=-1, provided n>0. [See Art. 334.] If x=1, the series becomes 1-+-..., which is convergent by Theorem V. a If x=1, the series becomes - (1 + 1 + } + ...), which is known to be divergent. [Art. 270.] 275. The condition for the convergency of the product of an infinite number of factors, and also some other theorems in convergency, will be proved in a subsequent chapter. The two important theorems which follow cannot however be deferred. be both convergent, and the third series 2 n ... +......+(uov2+ U2 v 2 - 1 + ......+U„vo) x2 + be formed, in which the coefficient of any power of x is the same as in the product of the two first series; then P will be a convergent series equal to UV, provided (1) that the series U and V have all their terms positive, or (2) that the series U and V would not lose their convergency if the signs were all made positive*. * This Article, and in fact the whole of this Chapter, is taken with slight modifications from Cauchy's Analyse Algébrique. First, suppose that all the terms in U and V are positive. 2n = Then Un Van Pan+terms containing a2 and x higher powers of x. Hence Un V2n> P2 2n 2n Hence P is intermediate to Un × Vn and U„„ × V, 2n 2n 2n 2n 2n' Now, the series U and V being convergent, U and Un both approach indefinitely near to U, also V and V, both approach indefinitely near to V, when n is indefinitely increased. Hence U × V and Un × Vn, and therefore also P, which is intermediate to them, will in the limit be equal to Ux V. Hence, when all the terms are positive, P = U× V. 2n 2n 2n Next, let the signs in the two series be not all positive, and let U' and V' be the series obtained by making all the signs positive in U and V; and let P' be the series formed from U and V' in the same way as P is formed from U and V. 2n 2n 2n 2n 2n' 2n Then UV-P cannot be numerically greater than U' x V' -P', for the terms in the latter expression are the same as those in the former but with all the signs positive. × 2n 2n Now, provided the series U and V do not lose their convergency when the signs of all the terms are made positive, it follows from the first case that U2 × V'2n − P's and therefore also UX V-P3 which is not numerically greater, must diminish indefinitely when n is increased without limit. Hence the limit of P is equal to the limit of UX V; so that P must be a convergent series equal to the product of U and V. 2n 2n If the series U and V are convergent, but are such that either of them would lose its convergency by making the signs of all its terms positive, the series P may or may not be convergent; and, when P is not convergent, the relation U × V = P does not hold good, for P has no definite value and cannot therefore be equal to U× V, although the coefficient of any particular power of x in the series P is always equal to the coefficient of the same power of x in the product of the series U and V. = 1' be equal to one another for all values of x for which they are convergent; then will a, b, α1 = b1, α = b1, &c. a2 For if the series are both convergent, their difference will be convergent. Hence a−b + (a−b1) x + (α, − b2) x2 + ...... = 0.........(i), for all values of a for which the series is convergent. x = The last series is clearly convergent when x = 0; and putting x=0 we have a,b,-0. Hence ab1. We now have = x {α-b1+(α-b2) x + (α, − b2) x2 + ......} = 0......(ii). Now for any value x, for which the series in (i) is convergent, a, b2+ (ɑ3 − b2) x2+ is equal to a finite limit, L, suppose. ...... 1 Hence (ii) may be written x, {a, - b1 + x1 L1} = 0; and, since this is true for all values of x,, however small, it follows that a1 - b1, must be numerically indefinitely small compared with L; that is, a1-b1 must be zero. It can now be proved in a similar manner that a, - b2 = 0, а3 - b1 = 0, &c. Hence if two series which contain x be equal to one another for all values of x for which the series are convergent, we may equate the coefficients of the same powers of x in the two series. The particular case of two series which have a finite number of terms was proved in Art. 91. |