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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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This entry was posted on February 4, 2010 at 4:38 pm and is filed under Holiday-related Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

4 Responses to “Things that equal e”

  1. Parviz Fekrat Says:
    February 4, 2010 at 10:43 pm | Reply

    Might as well include:

    i(pie)
    e = -1

  2. Ξ Says:
    February 5, 2010 at 8:14 am | Reply

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

  3. TwoPi Says:
    February 5, 2010 at 3:11 pm | Reply

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

  4. Head’s Up for e-Day « Let's Play Math! Says:
    February 5, 2010 at 5:15 pm | Reply

    […] Things that equal e […]

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