Sum 1 follow-up

Calculus Level 3

This problem is a follow-up to "Sum 1", my previous problem; if you've read my solution to "Sum 1", this problem should be straightforward.

Evaluate:

$\sum_{n=1}^{\infty} {\left(\frac{\left(-1\right)^n}{n}\sum_{k=1}^n {\left(\frac{\left(-1\right)^k}{k}\right)}\right)}$

The answer can be expressed as $$\dfrac{\pi^{a_1}}{6a_2}+\dfrac{\left(\ln{a_3}\right)^{a_4}}{a_5}$$, where $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$, and $$a_5$$ are prime positive integers; find $$10000a_1+1000a_2+100a_3+10a_4+a_5$$.

×