Consider the series
$$\sum{\frac{i^{n}}{n}}$$
We know that this series is not absolutely convergent as $\sum{|z_{n}|}$ gives harmonic series which is divergent.
However this series could be conditionally convergent or divergent.
I applied Ratio test (and got $|\frac{z_{n + 1}}{z_{n}}|$ approaches $1$) and Root test (which gave $\root{n}\of{|z_{n}|} = {(\frac{1}{n})}^{\frac{1}{n}}$ which is an indeterminate form as n approaches infinity. On manually checking for large values of n this approaches 1 though) but I am unable to certainly find an answer.
Another observation is that Root Test and Ratio test take mod and hence the result of these tests will not be different from what we will obtain for harmonic series. How will we then check if a series is conditionally convergent or not using these tests. Are these tests only to check absolute convergence?
Best Answer
The series is convergent by Dirichlet's Test, since $\;\sum\limits_{n=1}^\infty i^n\;$ is a bounded series and $\;\left\{\frac1n\right\}\;$ is a decreasing sequence convergent to zero.