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

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

• Does the sequence of partial sums converge? You may have to look at a few -- say, 4 -- subsequences. If they all converge, and to the same limit, then... Nov 7, 2016 at 21:31
• @ClementC. $4$? Isn't $2$ better? Nov 7, 2016 at 21:32
• Whatever works, I'd have gone for 4 (out of safety). Nov 7, 2016 at 21:33

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