Determine the radius of convergence of ∑ (-1)ⁿ(z+1)ⁿ / (n² + 1)

by RoRi

March 31, 2016

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

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

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

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

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

Thus, the series converges for $z$ such that $|z+1| < 1$ ; hence, $r = 1$ . Furthermore, the series converges on the boundary $|z+1| = 1$ by comparison with the series $\sum \frac{1}{n^2}$ .