Sum of Fractions in a Series

For a series of fractions where the denominators are products of consecutive integers, like $\frac{a}{b(b+1)} + \frac{a}{(b+1)(b+2)} + \dots$, there's a simple trick to find the sum.

Strategy: Add up all the numerators. Then, divide by the product of the smallest factor from the first denominator and the largest factor from the last denominator.

For example, $\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}$ is $\frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \frac{1}{5 \cdot 6}$. The sum of numerators is 4. The smallest factor is 2 and the largest is 6. The result is $\frac{4}{2 \cdot 6} = \frac{4}{12} = \frac{1}{3}$.