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.

  • 2
    $\begingroup$ Are you familiar with convergence tests for real series? Almost all the tests for real series also hold for complex series. $\endgroup$
    – Ulrik
    Jan 27 '14 at 21:42
  • $\begingroup$ Simply apply them to the series in question here, and you will get convergence for a disk about some center. $\endgroup$
    – Ayesha
    Jan 27 '14 at 21:43
  • $\begingroup$ 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. $\endgroup$
    – J.R.
    Jan 27 '14 at 21:44
  • $\begingroup$ I have no problem testing real series on convergence. I'll try again. $\endgroup$
    – 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.

The second sum is absolutely convergent for similar reasons.

  • $\begingroup$ It looks like you had mean $i^n = (-1)^{n+1}i$. But it is not true. $\endgroup$ Dec 30 '14 at 18:20

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