What is an example of a telescoping series and how do you find its sum?

1 Answer
Mar 8, 2015

A telescopic serie is a serie which can be written

#sum_{k=0}^n (a_{k+1}-a_k)#

This sum is equal to #a_{n+1}-a_0# because

#sum_{k=0}^n (a_{k+1}-a_k) = (a_1-a_0) + (a_2-a_1) + \ldots + (a_{n+1}-a_n)#.

An easy example is #sum_{k=1}^infty 1/(n(n+1))#.

Remark that #1/(n(n+1)) = 1/n - 1/(n+1)#, so,

#sum_{k=1}^N 1/(n(n+1)) = (1-1/2) + (1/2 - 1/3) + \ldots + (1/N - 1/(N+1))#

#sum_{k=1}^N 1/(n(n+1)) = 1 - 1/(N+1)#.

When #N -> +infty#, you get #sum_{k=1}^infty 1/(n(n+1))=1#.