A First Course in Complex Analysis · Problem 7.5.2

Taylor and Laurent Series — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · series-convergence-divergence

Problem

Determine whether each of the following series converges or diverges. (a)

\displaystyle\sum_{n\geq 1} \left(\frac{1+i}{\sqrt{3}}\right)^n

(b)

\displaystyle\sum_{n\geq 1} n\left(\frac{1}{i}\right)^n

(c)

\displaystyle\sum_{n\geq 1} \left(\frac{1+2i}{\sqrt{5}}\right)^n

(d)

\displaystyle\sum_{n\geq 1} \frac{1}{n^3 + i^n}