Do you know any almost identities?

Question

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?

Answer 1

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.

Answer 2

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.)

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

Answer 3

$$\pi\approx\dfrac{\ln(640320^3+744)}{\sqrt{163}}$$

Answer 4

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

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