---
tag: 8CC0635DY9
kind: proof
name: Proof that the exponential series has infinite radius of convergence
msc: [30B10, 33B10]
proves: 0PVJRACB9H
uses: [0VP56B4EMX, 0660QX1VPR, 0JBMYAA9QH, 0WFCGVMRRC, 0A74SX55QP, 0BWXV7GADW, 0W3K3QPEWT]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_function
    locator: Power series
---

## Proof

We use the ratio formula for the radius of convergence.

1. The series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is a [[power series|0VP56B4EMX]] with center $0$ and coefficients $a_n = \frac{1}{n!}$. Each $a_n$ is nonzero.
2. By the definition of the [[factorial|0WFCGVMRRC]], $(n+1)! = (n+1) \cdot n!$. So $\frac{\abs{a_{n+1}}}{\abs{a_n}} = \frac{n!}{(n+1)!} = \frac{1}{n+1}$.
3. We show $\frac{1}{n+1} \to 0$. Let $\varepsilon > 0$. By the [[Archimedean property|0A74SX55QP]], there is a natural number $N$ with $N > 1/\varepsilon$. For $n \ge N$, $0 < \frac{1}{n+1} < \frac{1}{N} < \varepsilon$.
4. By steps 2 and 3, the hypotheses of the [[ratio formula|0JBMYAA9QH]] hold with $L = 0$. So the [[radius of convergence|0660QX1VPR]] is $R = 1/0 = \infty$.
5. Let $x$ be real, and let $r = \abs{x} + 1$. Then $0 < r < R$. By the result on [[uniform convergence inside the radius|0BWXV7GADW]], the series converges absolutely on $[-r, r]$.
6. Since $x \in [-r, r]$, the series [[converges absolutely|0W3K3QPEWT]] at $x$. This holds for every real $x$.

## Idea

Each coefficient is the previous one divided by $n + 1$. So the coefficients shrink faster than any geometric sequence, and the ratio formula gives an infinite radius. Inside the radius a power series converges absolutely, and here the radius covers every real number.