Infinite series for $e$...

Question

How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this...

Answer 1

First prove by the ratio test that the series $\displaystyle \sum_{k=0}^{\infty} \frac{1}{k!}$ converges and denote the sum by $S$.

Then note that

$$\begin{align*} \left( 1 + \frac{1}{n} \right)^n & = \sum_{k=0}^n \binom{n}{k} \cdot 1^{n-k} \cdot \left( \frac{1}{n} \right)^k = \sum_{k=0}^n \frac{n (n-1) \ldots (n-k+1)}{k!} \cdot \frac{1}{n^k} \\[1ex] & = \sum_{k=0}^n \frac{1}{k!} \cdot \left( 1 - \frac{1}{n} \right)\left(1-\frac{2}{n}\right) \ldots \left( 1 - \frac{k-1}{n} \right) = \sum_{k=0}^n \frac{1}{k!} \cdot P_n^{(k)} \end{align*}$$

where

$$P_n^{(k)} = \left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right) \ldots \left(1-\frac{k-1}{n}\right).$$

We can see that for each $k \in \mathbb{N}$ we have $0 \leqslant P_n^{(k)} \leqslant 1$ and $\displaystyle \lim_{n \to \infty} P_n^{(k)} = 1$.

Fix $\varepsilon > 0$ and let $K$ be so large that $\displaystyle \sum_{k=0}^K \frac{1}{k!} \geqslant (1-\varepsilon)S$. Now let $N$ be so large that whenever $n \geqslant N$, for $k = 0, 1, \ldots, K$ we have $P_n^{(k)} > 1-\varepsilon$.

So for $n \geqslant N$

$$\begin{align*} (1-\varepsilon)^2 S \leqslant (1-\varepsilon) \sum_{k=0}^K \frac{1}{k!} \leqslant \sum_{k=0}^K \frac{1}{k!} P_n^{(k)} \leqslant \left(1+\frac{1}{n}\right)^n \leqslant \sum_{k=0}^n \frac{1}{k!} \leqslant S. \end{align*}$$

Therefore

$$e = \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n = S = \sum_{k=0}^{\infty} \frac{1}{k!}.$$

Answer 2

For me during calculus, the steps were:

1. Define $e$ as $\lim_{n\to\infty}\left(1+\frac1n\right)^n$
2. Define $\ln x$ as the inverse function to $e^x$.
3. Prove that $\frac{d}{dx} \ln x = \frac1x$
4. From 3, prove that $\frac{d}{dx}e^x=e^x$
5. Prove that, if a function $f$ is infinitly differentiable, then $$f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$ plus an error term which goes to $0$ in a lot of cases (including $e^x$).
6. From $5$, conclude that $$e^{x}=\sum_{n=0}^\infty\frac{x^n}{n!}$$
7. Plug in $x=1$ into (6).