Is it true that \[\sum_{k=0}^\infty \frac{k!!}{k!} = 1 + \sum_{k=0}^\infty \frac{1}{k!!}\]? Yes/probably.
Also \[\frac{1}{2}+\frac{1}{6}+\frac{1}{30}+\frac{1}{210}+\frac{1}{2310}+\frac{1}{30030}...=0.705230171791801... \\ \frac{1}{1}+\frac{1}{2}+\frac{1}{6}+\frac{1}{30}+\frac{1}{210}+\frac{1}{2310}+\frac{1}{30030}...=1.705230171791801...\]
oeis.org/A002110
Is \[\sum_{i=-\infty}^{\infty} f(x_i)\prod_{j=-\infty}^\infty\frac{x-x_j}{x_i-x_j} = cos(x)\] For \(f(x_i)=(-1)^i\), \(x_i=i\pi\) ?
We then have \[\sum_{i=-\infty}^{\infty} (-1)^i\prod_{j=-\infty}^\infty\frac{x-j\pi}{i\pi-j\pi} = cos(x)\]
If \[\sum_{n=0}^\infty n = \frac{-1}{12}\]
Does \[\prod_{n=0}^\infty x^n = \frac{1}{\sqrt[12]{x}}\]
Is there something special about \(12\) then...