For example, finding the smallest interval of convergence for given series: $\sum_{k=0}^{\infty} a_k (x-5)^k$ where for $x=9$ is known to be convergent and for $x=10$ divergent. Also, just for the sake of conceptualizing things, please include the largest possible interval as well.
I know how to find the interval of convergence, radius of convergence, and largest open interval for a series with knowns, but I'm not so sure about the unknowns. For this question, I'm thinking that the largest interval of convergence will be $x\in (0, 10)$ and that the smallest would be $(1,9)$. Also, how do I know if I should include the bounds for a series with unknowns like this one. Please help. Thanks in advance.