Chck if the series $\sum_{n\ge1}{\frac{i^n}{n}}$ absolutely convergent, semi convergent or divergent?

Question

Is the next series absolutely convergent, semi convergent or divergent? $$\sum_{n\ge1}{\frac{i^n}{n}}$$ where $i$ is the complex number such that $i^2=-1$. I don't know that I can use Leibniz Criterion because there is no $(-1)^n$, it's a complex noumber instead. Someone told me that I should use Dirichlet's test but I don't realy know how. Can somebody put me on the right track?

Answer 1

Hint:

• A series $$\sum_{n=1}^\infty (a_n + b_n i)$$ where $a_n, b_n\in\mathbb R$ is convergent if and only if both the series $$\sum_{n=1}^\infty a_n, \sum_{n=1}^\infty b_n$$ are convergent.
• In your case, $|a_n| = \frac1n$ if $n$ is even and $0$ if it is odd, while $|b_n|=\frac1n$ if $n$ is odd and $0$ if it is even. Also, look at the signs of $a_n$ and $b_n$.
• The series is absolutely convergent if and only if the series $$\sum_{n=1}^\infty\left|\frac{i^n}{n}\right|$$ is convergent.
• $|i^n| = |i|^n = 1$

Answer 2

Hint: Absoute convergence is trivially wrong. To prove convergence separate odd and even terms.