Power series convergence at specific x values

Question

for the series of $\sum C_n(x-2)^n$. We know that it converges when $x=4$ and diverges when $x=6$. lets say when $x=7, 5$ or $0.5$, how do we know if the series converges or not then?

Answer 1

A power series $\sum c_n(x-a)^n$ always has a radius of convergence $r\geq0$ so that the series converges when $|x-a|<r$ and diverges when $|x-a|>r.$ It may converge or diverge at the points where $|x-a|=r$, and it may converge at one of them and diverge at the other.

In this case, we have have $a=2$. We are told that the series converges at $x=4$ so we know $r\geq2$. We are told that it diverges when $x=6$ so we know $r\leq4$. Then the series certainly converges at $x=.5$ and diverges at $x=7$, but we don't know what happens at $x=5$. That will depend on the actual values of the $c_n$.