Sequences and Series 213
00
(b) The radius of convergence of L anbnzn is R;::: R1R2.
0
21. Show that :L:=l (1/n^2 )zn converges at every point of lzl = 1.
- Show that the series :L;'° zn! diverges at every point of the circle lzl = 1.
- Show that the series :L~ z^2 n diverges at every point of lzl = 1.
- Prove Corollary 4.4.
- Use the characterization of uniform continuity in Exercises 3.2, prob-
lem 8, to show that if a sequence of uniformly continuous functions
from a metric space ( X, d) into (Y, d') converges uniformly on X, the
limit function is uniformly continuous on X.
*26. Suppose that (1) :L;'° an converges, (2) an > 0, and (3) an ;::: an+I for
all n. Prove that nan ----7 0 as n ----7 oo.
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