Proof that $\sum_{k=1}^{\infty}\frac{i^k}{k}$ converges [duplicate]

determine whether the series convergence

$\sum _{n=1}^{\infty \:}\frac{i^n}{n} $

My teacher said it is convergent but the ratio test is inconclusive and the root test is inconclusive

The series is convergent but not absolutely convergent. Absolute convergence would say that the series $$ \sum_1^\infty \left| \frac{i^n}{n} \right| = \sum_1^\infty \frac{1}{n} $$ converges, and we know that is not the case.

However, we can break the series in question up as $$ \sum_{n=1}^\infty \frac{i^n}{n} = \sum_{m=1}^\infty (-1)^m \frac{1}{2m} + i \sum_{m=0}^\infty (-1)^m \frac{1}{2m+1} $$ and each of those alternating sign series can be shown to converge by grouping two terms together, getting a sum like $$ \sum_{m=1}^\infty (-1)^m \frac{1}{2m} = \frac12 \sum_{k=1}^\infty \left(\frac{1}{k} - \frac{1}{k+1} \right) = \frac12 \sum_{k=1}^\infty \frac1{k^2+k} $$ which converges by the ratio test.