SUmmation of natural logarithm

Question

Good day! Is there a formula that approximate the summation of natural logarithm of N as N runs from 1 to infinity?

Answer 1

The sum from 1 to n is the log of n!. Look up Stirling's formula.

Answer 2

If you mean $\log(1)+\cdots+\log(N)$, then

\begin{align} \sum_{i=1}^N\log(i)=\log(\Gamma(N+1)) \end{align}