Series convergence of $\frac{(-1)^n}{x^{2n+1}}$

Question

Does this series converge, and if so how would I prove it? I thought of using the ratio test but I'm not sure. The series is $$ \sum_{n=0}^\infty\frac{(-1)^n}{x^{2n+1}}. $$

Answer 1

Provided that $|x|>1$, the series converges. Note that $$\sum_{n=1}^\infty\frac{(-1)^n}{x^{2n+1}}=\frac1x\sum_{n=1}^\infty\left(\frac{-1}{x^2}\right)^n=\frac1x\frac1{1+\frac1{x^2}}=\frac{x}{x^2+1}$$