Power series centered at $x =0$

Question

I have this question in my advanced calculus textbook.

Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and diverges otherwise.

I came up with $$\sum \frac{(-1)^k}{2^kk} x^k$$ but this converges at the right endpoint and not all the endpoints. Any suggestions?

Answer 1

Bump the exponent up again, so you get $$ \sum_{k=1}^\infty \frac {(-1)^k}{k}\cdot\frac{x^{2k}}{4^k}. $$ Then at $x=-2$, the latter term vanishes like you want.