Can the endpoints of the domain of convergence of power series be absolute convergent at one end and divergent/conditionally convergent at the other

Question

Suppose $$|x-x_{0}|\leq\frac{1}{ \limsup\sqrt[n]{|a_n|}}$$ for the power series

$\sum\limits_{n=0}^{\infty} a_{n} (x-x_{0})^n$

My question is this, say if $x_a= x_{0}-\frac{1}{ \limsup\sqrt[n]{|a_n|}}$ is absolutely convergent, then can $x_b= x_{0}+\frac{1}{\limsup\sqrt[n]{|a_n|}}$ be divergent or conditionally convergent?

Answer 1

Let $ r= \frac{1}{\lim \sup\sqrt[n]{|a_n|}}$. Then we have

$\sum\limits_{n=0}^{\infty} a_{n} (x_a-x_{0})^n= \sum\limits_{n=0}^{\infty} a_{n}(-1)^nr^n$

and

$\sum\limits_{n=0}^{\infty} a_{n} (x_b-x_{0})^n= \sum\limits_{n=0}^{\infty} a_{n}r^n$.

Then we have:

$\sum\limits_{n=0}^{\infty} a_{n} (x_a-x_{0})^n$ converges absolutely $ \iff \sum\limits_{n=0}^{\infty} a_{n} (x_a-x_{0})^n$ converges absolutely .