146 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIESNote that, in applying either Theorem 4.13 or 4.14, if L = 1, the conver-
gence or divergence of the series is unknown, and further analysis is required to
determine the true state of affairs.
-------~EXERCISES FOR SECTION 4.3
- Evaluate
(a) E !It~>"·
n = O(b) f (2l,)".
n = O- Show that f <•;~>" converges for all values of z in the disk D2 ( -i) = (z: lz + ii < 2}
n=O
and diverges if lz + ii > 2. - Is the series f <4,!l" convergent? Why or why not?
n = O - Use the ratio test to show that the following series converge.
00
(a) I: ( .lfi )".
n = O(b) ~ L (Hi)" n2n ·
n = l(c) f^0 ~f ·
n = l
00 f l+i)2n(d) I: 2'•+1)!.
n = O- Use the ratio test to find a disk in which the following series converge.
00
(a) I: (1 + i)" z".
n = O
00 n(b) I: (3.:41)~.
n=O