Is it possible for a power series to be conditionally convergent at two different points?

Question

Like I stated in the title, I was just wondering if it's possible for a power series to be conditionally convergent at two different points. Are there any examples of power series that fit this criteria?

Any help is appreciated!

Answer 1

Try $$\sum_{n=1}^\infty \frac{(-1)^n x^{2n}}{n}$$ at $x=1$ and $-1$.

Answer 2

If the argument is complex, the answer is yes. For example $\sum\limits_{k=0}^\infty \frac{z^k}{k}$ has a radius of convergence $r=1$ and diverges for $z=1$, but converges for $z=-1$ and $z=\pm i$.

Wild guess: it converges for all points on the unit circle, except $z=1$