Working with precision 500 decimal digits, mpmath in sage computes:
$$\sum_{n=2}^\infty (-1)^n\log(n)=\log(1/2 \sqrt{2} \sqrt{\pi}) \qquad (1)$$
We believe the LHS of (1) diverges, so this isn't true.
Q1 Are there theoretical reasons mpmath to compute (1)?
