suppose $\sum a_n(x-3)^n$ is a power series that converges at $x=5$ and diverges at $x=0$, will $\sum a_n(x-3)^n$ converge at $x=8$ and at $x=6$?
I know that $2 \leq R \leq 3$ and that both 8, and 6 are outside the interval of convergence given that it is centered at 3, so the series will diverge at those points. Is this correct?
You are correct about $8$, but not about $6$. With the information that you have provided, nothing can be asserted about the convergence at $6$. Note that the distance from $6$ to $3$ is equal to the distance from $0$ to $3$.