Formula for Natural logarithm of $\pi$ Does any formula or expansion exist that gives $\ln \pi$ ?
The expansion should not just be any formula of $\pi$ with a $\ln$ before it. For example $\ln \pi$ = k + $\sum f(x)$ or something of this type.

One could use this formula to get
$$\ln(\pi)=\gamma+\sum_{n=1}^\infty\left[2\ln(n)-2\ln(n-0.5)-\frac1n\right]$$
Using the Euler product of the Riemann zeta function,
$$2\ln(\pi)=\ln(6)-\sum_{p\in\mathbb P}\ln(1-p^{-2})$$
where $p$ is prime.
Another interesting series:
$$\ln(\pi)=\ln(2)+2\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)$$
