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)?

[online code](https://sagecell.sagemath.org/?z=eJxVjkEOwiAQRfck3IElkLYWEjc23MKdcYG2WpICIwwLby-JmNTdvPf_T8Z5iAmZB29xpQTSYo7jOH158DDMkE21O1MJ8W3OqSyU5KJMi0Iunm_W32bLwokx3ishZZAt3uKTB9FddNeEC4-roAS12TXUQf8G-ZWQa_GH7QYn6rLBfY3A6x89avEB80BGGA==&lang=sage&interacts=eJyLjgUAARUAuQ==)