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Recently, I've read an article about almost identities and was fascinated. Especially astonishing to me were for example $\frac{5\varphi e}{7\pi}=1.0000097$ and $$\ln(2)\sum_{k=-\infty}^{\infty}\frac{1}{\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)^k}=\pi+5.3\cdot10^{-12}$$ So I thought it would be nice to see a few more. Therefore, my question is: Do you know a fascinating almost identity? Can you, in some sense, prove it?

Eric Stucky
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Redundant Aunt
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    The first one really looks like an unremarkable coincidence. Of all the arithmetic combinations of $\pi$, $e$, $\varphi$, and small primes, it isn't surprising that some will be fairly close to integers. But maybe I'm missing something. – Jonas Meyer Feb 02 '15 at 20:15
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    I know this one, but it's a complete identity (not an "almost" one): $\sum\limits_{n=1}^{\infty}\frac{1}{2^n}=\sum\limits_{n=1}^{\infty}\frac{1}{2^n \ln(2^n)}$ – barak manos Feb 02 '15 at 20:17
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    http://xkcd.com/1047/ this xkcd has loads – Mmm Feb 02 '15 at 20:19
  • The infinite series diverges. Typo, perhaps ? – Lucian Aug 22 '19 at 16:08

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A lot of examples are found when you seek almost-rationals - Ramanujan was an expert in that.

http://en.wikipedia.org/wiki/Almost_integer

These are tempting to just identify with $\pi/2$ until the pattern breaks down unexpectedly:

http://en.wikipedia.org/wiki/Borwein_integral

One that fascinates me is $\gamma\sim e^{-\gamma}\sim W(1)$ where $W$ is the Lambert function and $\gamma$ is Euler-Mascheroni constant.

orion
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There is a substantial list on Wikipedia:

http://en.wikipedia.org/wiki/Mathematical_coincidence

Some of the more interesting (imo) examples are: \begin{equation} \pi\approx\frac{4}{\sqrt{\varphi}}\\ \pi^4+\pi^5\approx e^6\\ \frac{\pi^{(3^2)}}{e^{(2^3)}}\approx10\\ e^{\pi}-\pi\approx20 \end{equation} There's also the claim made in the April 1975 Scientific American (more specifically, the April Fool's claim) that Ramanujan had predicted that $e^{\pi\sqrt{163}}$ is an integer. (It isn't, but it is extremely close.)

Oh, and lest we forget, $\pi=3.2$.

KSmarts
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    By the way, the fact that $e^{\pi\sqrt{163}}$ is very close to an integer is not a coincidence. It comes from the theory of modular forms, precisely from computing the $j$-invariant for a quadratic extension of $\mathbb{Q}$ with class number one. – Jack D'Aurizio Feb 02 '15 at 22:16
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$$\pi\approx\dfrac{\ln(640320^3+744)}{\sqrt{163}}$$

Lucian
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You can produce a lot of those with the following tool by Robert Monafo:

http://mrob.com/pub/ries/

flawr
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