Let $f$ be an entire function. Prove Taylor's theorem, that there exist complex numbers $c_{0}, c_{1}, \ldots$ such that $f(z)=\sum_{n=0}^{\infty} c_{n} z^{n}$ for all $z$. [You may assume Cauchy's Integral Formula.]

For a positive real $r$, let $M_{r}=\sup \{|f(z)|:|z|=r\}$. Explain why we have

$$\left|c_{n}\right| \leqslant \frac{M_{r}}{r^{n}}$$

for all $n$.

Now let $n$ and $r$ be fixed. For which entire functions $f$ do we have $\left|c_{n}\right|=\frac{M_{r}}{r^{n}} ?$