Determine the radius of convergence of ∑ (z+3)ⁿ / (n+1)2ⁿ

by RoRi

March 31, 2016

Determine the radius of convergence $r$ of the power series:

$\sum_{n=0}^{\infty} \frac{(z+3)^n}{(n+1)2^n}.$

Test for convergence at the boundary points if $r$ is finite.

Let $a_n = \frac{(z+3)^n}{(n+1)2^n}$ . Then

$\begin{align*} \lim_{n \to \infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to \infty} \left( \frac{(z+3)^{n+1}}{(n+2)2^{n+1}} \right) \left( \frac{(n+1)2^n}{(z+3)^n} \right) \\[9pt] &= \lim_{n \to \infty} \frac{(z+3)(n+1)}{2(n+2)}. \end{align*}$

This implies the series converges for $|z+3| < 2$ which implies $r = 2$ . Furthermore, the series converges on the boundary except the point $z = -1$ . Hence, the series converges for all $z$ such that $|z+3| \leq 3$ and $z \neq -1$ .