Convergency of the power series at two points Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ & divergent at $2-3i$. (b) convergent at $2-3i$ & divergent at $-2+5i$. (c) convergent at both. (d) divergent at both.

The series is a power series centered at $i-3$.

It has a radius of convergence, unique by the theory. Inside that, the series converges absolutely, outside it does not.

From the data you have (observe that $5i$ and $-3i$ have the same distance from the center $i-3$) you can infer that the radius of convergence is 5 (the distance between the center and any of these two points): in fact the radius has to be bigger than 5 since the series converges at $5i$, but it has to be smaller than 5 since we have divergence at $-3i$.

At this point, the point $-2+5i$ is inside the radius of convergence, while $2-3i$ is not.