Advanced Calculus fi..

424 Advanced Calculus, Fifth EditionThe series converges absolutely whenI X - al el, thatis, Ix-al r*.If r* = 0, the test shows divergence except for x = a. If r* = oo, so thatthe test shows that the series converges absolutely for all x.The case in which the limit (6.39) exists is treated in the same way by the roottest. This case is included in the final formula (6.40), which follows at once fromTheorem 21(a) of Section 6.8, forThe series converges absolutely when the upper limit is less than 1, that is, whenI X -a1 e - 1 = r*,limn+mand diverges when Ix - a 1 > r*.Thus however it may be determined, there is a number r*, 0 5 r* 5 oo, with theproperties described. It remains to prove that if 0 < rl < r*, then the series convergesuniformly for Ix - a1 5 rl. This follows from the M-test, with Mn = Icnlr;, forand this converges, since Ixl - a 1= rl < r*. For Ix - a 1 5 rl,so that the convergence is uniform. The main idea of this proof is emphasized inSection 6.12-the slowest convergence of a power series occurs toward the ends ofthe interval of convergence.It should be emphasized that when r* is a finite positive number, the series mayconverge or diverge for each of the end-values x = a + r*, x = a - r*. These mustbe investigated separately for each series.EXAMPLE 1similarly, (6.39) givesxEl 5. Here cn = 1 /n2, and (6.38) gives(n +r* = lim - = 1:" %