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Elementary Functions — Howell & Matthews
Mathematics — Complex Analysis · ratio-test-convergence
Problem Use the ratio test to show that the following series converge.
((a)) ∑ n = 0 ∞ ( 1 + i 2 ) n \displaystyle\sum_{n=0}^{\infty} \left(\frac{1+i}{2}\right)^n n = 0 ∑ ∞ ( 2 1 + i ) n ∑ n = 1 ∞ ( 1 + i ) n n 2 n \displaystyle\sum_{n=1}^{\infty} \frac{(1+i)^n}{n2^n} n = 1 ∑ ∞ n 2 n ( 1 + i ) n ∑ n = 1 ∞ ( 1 + i ) n n ! \displaystyle\sum_{n=1}^{\infty} \frac{(1+i)^n}{n!} n = 1 ∑ ∞ n ! ( 1 + i ) n ∑ n = 0 ∞ ( 1 + i ) 2 n ( 2 n + 1 ) ! \displaystyle\sum_{n=0}^{\infty} \frac{(1+i)^{2n}}{(2n+1)!} n = 0 ∑ ∞ ( 2 n + 1 )! ( 1 + i ) 2 n Browse the problem library