Problem

Let $S=\{r:r \text{ is a positive number such that } \sum_{k=0}^{\infty}x^k/k! \text{ converges absolutely for } |x|<r\}$. Which of the following statement is true?

(a) $S$ is empty.

(b) $S$ is nonempty and $r\le 1$ for all $r\in S$.

(c) $S$ is nonempty and if for some $r\in S$, a function $f$ is defined as $f(x)=\sum_{k=0}^{\infty}x^k/k!$ for $x\in(-r,r)$, then $f'(x)>f(x)$ for $x\in(0,r)$.

(d) $S=(0,\infty)$.

(e) None of the above statements is true.