What is the value of the following infinite series?

\(\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}\)

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-15.

An alternative solution was submitted by 김호진 (2009학번, +2). His alternative solution did not check whether the swapping two infinite sums can be done.