1.5.3 Sum of Fractions Series

The best way to illustrate this trick is by example:

$$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} = \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \frac{1}{5 \cdot 6}$$ $$= \frac{1 + 1 + 1 + 1}{2 \cdot 6} = \frac{4}{12} = \frac{1}{3}$$

So the strategy when you see a series in the form of $\frac{a}{b \cdot (b+1)} + \frac{a}{(b+1) \cdot (b+2)} + \dots$ is to add up all the numerators and then divide it by the smallest factor in the denominators multiplied by the largest factor in the denominators.

Let’s look at another series:

$$\frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} = \frac{1}{6 \cdot 7} + \frac{1}{7 \cdot 8} + \frac{1}{8 \cdot 9} + \frac{1}{9 \cdot 10} + \frac{1}{10 \cdot 11}$$ $$= \frac{1 + 1 + 1 + 1 + 1}{6 \cdot 11} = \frac{5}{66}$$

Problem Set 1.5.3