I came across a problem that says:
It is given that $\sum_{n=0}^{\infty}a_{n}z^{n}$ converges at $z=3+4i.$
Then the radius of convergence of the power series $\sum_{n=0}^{\infty}a_{n}z^{n}$ is
(a)$\leq 5$
(b)$\geq 5$
(c)$<5$
(d)$>5$.
We know if a power series $\sum_{n=0}^{\infty}a_{n}z^{n}$ converges for $z=z_{0},$ then it is absolutely convergent for every $z=z_{1},$ when $|z_{1}|<|z_{0}|.$ Using this property, i can conclude that $(a)$ is the correct choice as equality sign occurs keeping in mind that the given series converges at $|3+4i|=5.$ Am i going in the right direction? Please help. Thanks in advance for your time.