determine whether the series convergence $\sum _{n=1}^{\infty \:}\frac{i^n}{n}$

Question

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

Answer 1

Hint: Subdivide the Series in two sub-series: one over $2n$ and one over $2n+1$ and show that these converge.

Answer 2

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.