Series that converges on $[-1,1]$

Question

What is an example of a series that converges only on $[-1,1]$? I am unable to come up with one right now for some reason. Thanks

Answer 1

$$\sum \frac{x^n}{n^2}$$

$$\ \ $$

Answer 2

One may consider

$$ \sum_{n=1}^\infty e^{-(1-|x|)n^2-n} $$

• If $x \in [-1,1]$ then $$\left| e^{-(1-|x|)n^2-n}\right|\leq e^{-n}$$ and the initial series converges by comparison.

• If $x \notin [-1,1]$ then, as $n \to \infty$, using
  $-(1-|x|)n^2-n=n((|x|-1)n-1)\to +\infty$, one gets $$ \lim_{n \to +\infty}e^{-(1-|x|)n^2-n}=+\infty \neq0$$ and the initial series diverges trivially.