Infinite sum of logarithms Is there any closed form for this expression
$$
\sum_{n=0}^\infty\ln(n+x)
$$

A simple way to see why it diverges for $x>0$ is as follows:
$$\sum_{n=0}^\infty \ln(n+x) \ge \sum_{n=1}^\infty\ln(1+x) = \infty$$
as an infinite sum of a constant positive argument is always infinite (I have used monotonicity of the logarithm here).
For $x\le 0$, $\ln(0+x)$ is not defined, hence the entire sum is not defined.
