Sequences & series

Many series become easy when each term can be rewritten as the difference of two simpler fractions. For a product of consecutive integers we have the identity \(\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}\). When you add many such terms, almost everything cancels – this is called a telescoping sum. After the cancellation only the first and the last pieces remain, so the whole sum is just \(\frac{1}{1} - \frac{1}{n+1}\) for a sum that runs from \(k=1\) to \(k=n\). This trick lets you evaluate long‑looking finite sums in a single line.