Another, well, interesting but not necessarily "challenging" problem from the series of problems that I started posting a few days ago. I hope you'll like it!

Suppose the integer $\displaystyle n \geq 1$ is given. Evaluate $\displaystyle I_n=\sum_{j_1 = 1}^{\infty} \sum_{j_2 = 1}^{\infty} \cdots \sum_{j_n = 1}^{\infty} \frac{1}{j_1 j_2 \cdots j_n(j_1 + j_2 + \cdots + j_n)}.$