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?

You can produce a lot of those with the following tool by Robert Monafo:
http://mrob.com/pub/ries/
