Sequences and Series, Telescoping Series

Telescoping Series

The terms of a telescoping series cancel, simplifying the process of computing the sum. Let's illustrate with an example.

Consider the series s_j = 1/(j²+3j+2). What is its sum? The terms of this series can be rewritten:

s_j = 1/(j+1) - 1/(j+2)

As we add terms together, intermediate fractions "telescope" away, and the j^th partial sum becomes 1/2 - 1/(j+2). The series converges to 1/2.