$\sum_{n=1}^{\infty} \frac{i^n}{n}$

Question

I want to show that this series converges $$\sum_{n=1}^{\infty} \frac{i^n}{n}.$$ It is easy to see that it is not absolutely convergent. No use looking at the radius of convergence either. Can Someone help me?

Answer 1

It converges by Dirichlet's test: the partial sumes of the series $\sum_{n=1}^\infty i^n$ are bounded, the sequence $\left(\frac1n\right)_{n\in\mathbb N}$ is monotonic and converges to $0$.

Answer 2

Hint

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