find an example for a series $a_{n}$ that satisfies the following:
$a_{n}\xrightarrow[n\to\infty]{}0$
${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges
There is a way to insert parentheses so ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ will converges.
I was thinking about the series:$ 1-1+\frac{1}{2}+\frac{1}{2}-\frac{1}{2}-\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}+...$
But I don't know how to prove 2.
Also will be nice to hear another examples, if any.