Do you know any almost identities? 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?
 A: 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.
A: 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, $\pi=3.2$.
A: $$\pi\approx\dfrac{\ln(640320^3+744)}{\sqrt{163}}$$
A: You can produce a lot of those with the following tool by Robert Monafo:
http://mrob.com/pub/ries/
