Advanced Calculus fi..

398 Advanced Calculus, Fifth EditionEXAMPLEIn the seriesthe odd terms and even terms each form a geometric series with ratio $. The ratiosof successive terms arein general the ratio of an even term to the preceding one is 5 and the ratio of an oddterm to the preceding one is i. The ratios approach no definite limit but are alwaysless than or equal to r = $. Accordingly, the series converges.It is of interest to note that part of Theorem 20 can be stated in terms of upperlimits (Section 6.3):THEOREM 20(a)If an # 0 for n = 1,2, . . . andithen the series C a, converges absolutely.For if the upper limit is k < 1, then the ratio must remain below k + 6, for nsufficiently large, for each given E > 0. One can choose 6 so that k + 6 = r < 1,and Theorem 20 applies.di.It should be remarked that no mclusions can be drawn from either of therelations: /- Qn+11 =l. ;.tl :>The conditionan+1- lim - > 1n-mI an Iimplies divergence, but this is not as precise as Theorem 20.The root test (Theorem 19) can be extended in the same way as the ratio test:THEOREM 21 If there exist a number r such that 0 < r < 1 and an integer N suchthatwr forn > N,then the series C,dO_, a, converges absolutely. If, on the other hand,t'for infinitely many values of n, then the series diverges.The proof in the case of convergence is the same as that of Theorem 19. If5 1 for infinitely many values of n, then la, I 2 1 for infinitely many valuesof n, so that the nth term cannot converge to 0, and the series diverges.