Basic Analysis I — Real Analysis · Problem 5.4.3

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · exponential-series

Problem

(requires §4.3) Use Taylor's theorem to study the remainder term and show that for all

x \in \mathbb{R}

e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.

Hint: Do not differentiate the series term by term (unless you would prove that it works).