Things that equal e

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e-day is coming up on Sunday, and I’ve already started making goodies to share on Friday (not wanting to fall into the trap of burning everything again).  Instead of writing “e” on the top, I’m thinking of putting in one of the following:

• $\displaystyle\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) ^n$
• $\displaystyle\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}}$
• $\displaystyle 1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...$
• The x-value for which $\displaystyle x^\frac{1}{x}$ is as large as possible.
• $\cosh{1}+\sinh{1}$
• $\cos{i}-i\sin{i}$
• $\displaystyle \frac{\sinh{\pi}}{\pi}+\frac{2\sinh{\pi}}{\pi}\cdot\sum_{n=1}^{\infty}\frac{(-1)^n}{1+n^2}\left(\cos{n}-n\sin{n} \right)$
• (from OEIS A001113) $\displaystyle \left(\frac{16}{31}\cdot\left(\sum_{n=1}^{\infty}\left(\frac{n^3+n+2}{2^{n+1}n!} \right)+1\right)\right)^2$
• (also from OEIS A003417) The number represented by the cool looking continued fraction [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, …]
$1+1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cfrac{1}{1+...}}}}}}}}}$

(Any other good ones?)

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4 Responses to “Things that equal e”

1. Parviz Fekrat Says:

Might as well include:

i(pie)
e = -1

2. Ξ Says:

I thought about that, but it equals -1. Oh, but I could use the (i*Pi)th root of (-1)!

3. TwoPi Says:

So that’s $(-1)^{1/i \pi}$, or equivalently $(-1)^{-i/\pi}$.

4. Head’s Up for e-Day « Let's Play Math! Says:

[…] Things that equal e […]