examples of conditionally convergent series other than alternating harmonic series

I've been trying to find interesting examples of conditionally convergent series but have been unsuccessful. I'd particularly like to find a conditionally convergent series of the following form:

$\sum_{n=1}^\infty a_n$ where $a_n=f(n,z)$ with $Im(z)≠0$

(or even better $a_n=f(n,z^n)$, with $Im(z)≠0$)

but if you know of any interesting conditionally convergent series at all, please share. Alternating harmonic series appears to be the most popular example everywhere.

• You can use $\sum \frac{(-1)^n}{n^\alpha}$ for any $0<\alpha\leq 1$. But I'm not sure what you mean by $f(n,z^n)$. Could you give an example of a sequence of that form? – Kitegi Nov 16 '15 at 18:34
• Alternating harmonic series is the easiest one to work with for many reasons. I would also suggest Kitegi's example before all others. However, it's very hard to rearrange those in a simple way and still get a convergent series (if this is something you are interested in). – Maxim Gilula May 28 at 0:58

2 Answers

In the following I assume that by $f(z)$ you mean an analytic function of $z$. Since $f(z)$ does not depend on $n$, there are no examples of the first type. The second type is a little more complicated to analyze. Since $a_n\to0$ is a necessary condition for convergence, we must have $\lim_{n\to\infty}f(z^n)=0$. If $|z|<1$ then $z^n\to0$. This means that $f$ has a zero at $z=0$. Then we have $|f(z)|\le C\,|z|$ for some constant $C$ and $\sum a_n$ converges absolutely. A similar reasoning aplies if $|z|>1$. The only case left is $z=e^{2\pi\alpha i}$ with $\alpha$ real. If $\alpha$ is rational, then $z^n$ has a finite number of values, and $a_n=f(z^n)=f(e^{2\pi\alpha ni})\to0$ can happen only if $a_n=0$ for all $n$. If $\alpha$ is irrational, then $\{e^{2\pi\alpha ni}:n\in\mathbb{N}\}$ is dense in the circle, and again the only possibility of convergence is $a_n=0$ for all $n$.

Now for some examples:

1. $\sum_{n=1}^\infty (-1)^nb_n$, where $b_n>0$ is decreasing and $\lim_{n\to\infty}b_n=0$.
2. $\displaystyle\sum_{n=1}^\infty\frac{\sin(n\,x)}{n}$, $0<x<\pi$.
3. $\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{n-(-1)^n}$.

$1 -1/2^2 -1/2^2+1/3^2+1/3^2+1/3^2 -1/4^2 -1/4^2-1/4^2-1/4^2 + \cdots$