Find the value of $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \dots + \frac{1}{9 \times 10}$.

Explanation

1. 01

   Each term splits: $\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1}$.

2. 02

   The sum becomes $\left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \dots + \left(\frac{1}{9} - \frac{1}{10}\right)$.

3. 03

   Everything in the middle cancels, leaving $1 - \frac{1}{10} = \frac{9}{10}$.