Sequence convergence and parentheses insertion

Question

find an example for a series $a_{n}$ that satisfies the following:

1. $a_{n}\xrightarrow[n\to\infty]{}0$

2. ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges

3. 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.

Answer 1

D'Alembert's ratio test will help you with nr. 2.