# What is a telescoping infinite series?

${\sum}_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{n + 1}\right)$
$= \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots$
As you can see above, terms are shifted with some overlapping terms, which reminds us of a telescope. In order to find the sum, we will its partial sum ${S}_{n}$ first.
${S}_{n} = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n + 1}\right)$
$= 1 - \frac{1}{n + 1}$
${\sum}_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{n + 1}\right) = {\lim}_{n \to \infty} {S}_{n} = {\lim}_{n \to \infty} \left(1 - \frac{1}{n + 1}\right) = 1$