Telescoping sum (${\sum }_{k=1}^n(a_k-a_{k-1})=a_n-a_0$)

A telescoping sum cancels intermediate terms, leaving boundary terms. For a finite sequence $a_0,\dots ,a_n$, the displayed identity follows because every $a_1,\dots ,a_{n-1}$ 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.