Absolute convergence of a power series.

Question

Is it possible to have a power series with radius of convergence $R $ such that there exist $z_1$ and $z_2$ satisfying $|z_1|=R$, $|z_2|=R$ whereas the power series absolutely converges at $z_1$ and diverges at $z_2$.

Answer 1

If you meant $\lvert z_1\rvert=R$ and $\lvert z_2\rvert=R$, then the answer is negative. If $\sum_{n=0}^\infty a_n(z-a)^n$ converges absolutely at $z_1$ and if $\lvert z_1-a\rvert=\lvert z_2-a\rvert$, then the series also converges absolutely at $z_2$. After all$$\sum_{n=0}^\infty\bigl\lvert a_n(z_1-a)^n\bigr\rvert=\sum_{n=0}^\infty\bigl\lvert a_n(z_2-a)^n\bigr\rvert.$$