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Elementary Functions — Howell & Matthews
Mathematics — Complex Analysis · ratio-test-radius-of-convergence
Problem Use the ratio test to find a disk in which the following series converge.
((a)) ∑ n = 0 ∞ ( 1 + i ) n z n \displaystyle\sum_{n=0}^{\infty} (1+i)^n z^n n = 0 ∑ ∞ ( 1 + i ) n z n ∑ n = 0 ∞ z n ( 3 + 4 i ) n \displaystyle\sum_{n=0}^{\infty} \frac{z^n}{(3+4i)^n} n = 0 ∑ ∞ ( 3 + 4 i ) n z n ∑ n = 0 ∞ ( z − i ) n ( 3 + 4 i ) n \displaystyle\sum_{n=0}^{\infty} \frac{(z-i)^n}{(3+4i)^n} n = 0 ∑ ∞ ( 3 + 4 i ) n ( z − i ) n ∑ n = 0 ∞ ( z − 3 − 4 i ) n 2 n \displaystyle\sum_{n=0}^{\infty} \frac{(z-3-4i)^n}{2^n} n = 0 ∑ ∞ 2 n ( z − 3 − 4 i ) n Browse the problem library