TITLE: Express $\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$ in terms of harmonic numbers
QUESTION [8 upvotes]: Express $$\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$$ in terms of the harmonic numbers $H_n$.

I guess that there could be several approaches for doing this.

REPLY [4 votes]: $\begin{array}\\
S
&=\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}\\
&=\sum_{j=1}^{n}\sum_{i=1}^{n} \frac1{j}(\frac1{i}-\frac1{i+j})\\
&=\sum_{j=1}^{n}\frac1{j}\sum_{i=1}^{n} (\frac1{i}-\frac1{i+j})\\
&=\sum_{j=1}^{n}\frac1{j}\left(\sum_{i=1}^{n} \frac1{i}-\sum_{i=1}^{n}\frac1{i+j}\right)\\
&=\sum_{j=1}^{n}\frac1{j}\sum_{i=1}^{n} \frac1{i}-\sum_{j=1}^{n}\frac1{j}\sum_{i=1}^{n}\frac1{i+j}\\
&=H_n^2-\sum_{i=1}^{n}\sum_{j=1}^{n}\frac1{j}\frac1{i+j}\\
&=H_n^2-S\\
\text{so}\\
S
&=\frac12 H_n^2\\
\end{array}
$