Checking for convergence of $\sum_{n=1}^\infty \frac{(-i)^n}{n}$
Solution 1:
Just apply Dirichlet's test and use the fact that, for each natural $N$,$$\left\lvert\sum_{n=1}^Ni^n\right\rvert=\left\lvert\frac{i-i^{N+1}}{1-i}\right\rvert\leqslant\sqrt2.$$