Telescoping sum
= Telescoping sum
{title2=$\sum_{k=1}^{n}(a_k-a_{k-1})=a_n-a_0$}
A <telescoping sum> cancels intermediate terms, leaving boundary terms. For a finite <sequence> $a_0,\ldots,a_n$, the displayed identity follows because every $a_1,\ldots,a_{n-1}$ occurs once with each sign. Applying it to an infinite <series> also requires checking the <limit> of the boundary terms; finite cancellation alone does not prove that the <series> is a <convergent series>.