Find the Unknown sum

The $n^\text{th}$ denominator is given by $n(n+7)$ (this is easy to find by looking at their differences).

Since $\frac{1}{n(n+7)}=\frac{1}{7n}-\frac{1}{7(n+7)}$

the sum telescopes: $\begin{aligned} \sum_{n=1}^\infty \frac{1}{n(n+7)}&=\sum_{n=1}^\infty \frac{1}{7n}-\sum_{n=1}^\infty \frac{1}{7(n+7)} \\ &=\sum_{n=1}^\infty \frac{1}{7n}-\sum_{n=8}^\infty \frac{1}{7n} \\ &=\sum_{n=1}^7 \frac{1}{7n} \\ &=\boxed{\frac{363}{980}}\end{aligned}$