Consequence of a Power Series Convergence at a Point

Question

My Elementry Differential Equations book by Boyce, (10th edition) claims the following about power series convergence:

"If the power series $\sum_{n=0}^{\infty} a_{n} (x-x_{0})^n$ converges at $x= x_{1}$, then it converges absoultely for $\left | x-x_{0} \right | < \left | x_{1}-x_{0} \right |$"

This statement makes sense if the series converges absolutely at $x = x_{1}$. But is the claim also true if the series converges conditionally at $x = x_{1}$?

Answer 1

Let $ R $ be the radius of convergence.

this means that if $ \sum a_n(x_1-x_0)^n $ converges, then

$$|x_1-x_0|\le R$$

So, $$|x-x_0|<|x_1-x_0| \implies$$ $$|x-x_0|<R \implies $$ $$\sum a_n (x-x_0)^n \; is \; absolutely\; convergent$$