Question

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

Answer 1

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`.