$f(z)=\sum_{n=1}^{\infty} a_n(z-z_0)^n$ such that $\sum_{n=0}^{\infty} f^n(a)$

Question

Let $f(z)=\sum_{n=1}^{\infty} a_n(z-a)^n$ such that $\sum_{n=0}^{\infty} f^n(a)$ converges then is it necessary that $f(z) $ is entire. I have tried this by giving counter example but all in vain , Any hints leading to answer are deeply appreciated

Answer 1

If $f(z)=\sum\limits_{k=1}^{\infty} a_n(z-a)^{n}$ in some neighborhood of $a$ then $a_n =\frac {f^{(n)} (a)} {n!}$. Since $f^{(n) }(a)\to 0$ it follows (by comparison with exponential series) that the series converges for all $z$ and $f$ extends to an entire function.