Advanced Calculus fi..

402 Advanced Calculus. Fifth Editionfor n 2 nl, thenlRnl i lan2+11 = T,*, n I nl; (6.19)Ian+] l - lan+2 lthe sequence Ti is monotone decreasing and converges to 0.ProoJ: The first statement follows from Theorem 22 and the fact that the ratio testis a comparison of the given series with a geometric series. Thus one has, as in theproof of Theorem 17,The second statement emphasizes the fact that if the test ratio converges to L,then the ratio cannot remain less than or equal to a number r less than L. Thus r 2 L.One can use r = L only whenfor n > N, so that the limit L is approached from below. ...$The third statement concerns the case in which the test ratios are steadily decreasingand hence approaching a limit L. One cannot use L in such a case; however,under the assumptions made, one can use r = lan+2/an+l for n 2 nl, so thatIThis gives formula (6.19).EXAMPLE 3 xz,, s. The test ratio is found to be.ifirThis converges to i but is always less than i. Hence we can use r = i and, forexample,EXAMPLE 4C,"=I" Here the test ratio isAgain the limit is 1, but the ratio is always greater than f . The successive ratios aredecreasing since. -