Got some $e$ ?

We all know by now one of these definition's of $e$ : $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \sum_{n=0}^\infty \frac{1}{n!}$ But of course, these aren't the only ways to express it. This discussion is about sharing our favorite definitions of $e$ and why we think they're interesting.

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$e=\lim_{n\to0}\sqrt[n]{1+n}$ This version is simply an application of $\lim_{n\to\infty}f(n)=\lim_{n\to0}f(1/n)$ , but it's probably my favorite one to look at:

$e=x:\frac{d}{dx}\sqrt[x]{x}=0$ From Wikipedia: "This maximum occurs precisely at x = e. For proof, the inequality $e^{y}\geq y+1$ , from above, evaluated at $y=(x-e)/e$ and simplifying gives $e^{x/e}\geq x$ . So $e^{1/e}\geq x^{1/x}$ for all positive x.

William Crabbe - 3 years ago

so what are you trying to say

Nahom Assefa - 2 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Got some eee?

Comments

Got some $e$ ?