## Posts Tagged ‘e-day’

### Things that equal e

February 4, 2010

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?)

The date was February 7, 2005. The time was 5:00am. I’d gotten up early to cook, and this year decided to try a recipe that involved baking the brownies, then putting white chocolate chips and crushed candy canes on top. I placed the whole pan under the broiler for a few minutes for the chips to soften before spreading them around. And I’m sure all would have gone well if TwoPi hadn’t asked from the dining room, “Hey, are there any nonzero integers a, b, and c such that $a^2+b^3=c^4$?” Naturally I started to work on that question, and forgot all about the brownies.