# How to test convergence of complex series?

I've been looking for examples of how complex series are tested on convergence, however I could not quite find what I wanted. So I'm asking here, how do I handle, for example:

$$\sum_{n=1}^{\infty} \frac{i^n}{n}$$ or $$\sum_{n=1}^{\infty} \frac{(2i)^k + 4k}{3^k + i^k}$$ ?

Thanks for the help.

• Are you familiar with convergence tests for real series? Almost all the tests for real series also hold for complex series. Jan 27 '14 at 21:42
• Simply apply them to the series in question here, and you will get convergence for a disk about some center. Jan 27 '14 at 21:43
• Use that i^n=\left\{\begin{align}1 & n\equiv 0,\,\text{(mod 4)}\\ i, & n\equiv 1\,\text{(mod 4)}\\ -1, & n\equiv 2\,\text{(mod 4)}\\ -i, & n\equiv 3\,\text{(mod 4)}\end{align}\right. and decompose into real and imaginary part.
– J.R.
Jan 27 '14 at 21:44
• I have no problem testing real series on convergence. I'll try again.
– Nhat
Jan 27 '14 at 21:44

Note that $i^2=-1$ and so the 1st series just becomes $\sum\limits_{n=1}^{\infty}\dfrac{(-1)^{n+1}i}{n}$ which converges by alternating series test.
• It looks like you had mean $i^n = (-1)^{n+1}i$. But it is not true. Dec 30 '14 at 18:20