Telescoping Series

A telescoping series does not have a set form, like the geometric and p-series do. A telescoping series is any series where nearly every term cancels with a preceeding or following term. For instance, the series

the sum over n from 1 to infinity of 1/n - 1/(n+1)

is telescoping. Look at the partial sums:

the sum over i from 1 to n of 1/i - 1/(i+1) = (1/1 - 1/2) + (1/2 - 1/3) + ... + (1/n - 1/(n+1))

= 1 - 1/(n+1)

because of cancellation of adjacent terms. So, the sum of the series, which is the limit of the partial sums, is 1.

You do have to be careful; not every telescoping series converges. Look at the following series:

the sum over n from 1 to infinity of n - (n + 1)

You might at first think that all of the terms will cancel, and you will be left with just 1 as the sum.. But take a look at the partial sums:

the sum over i from 1 to n of i - (i + 1) = (1 - 2) + (2 - 3) + ... + (n - (n + 1)) =

1 - (n + 1) = -n.

This sequence does not converge, so the sum does not converge. This can be more easily seen if you simplify the expression for the term. You find that

the sum over n from 1 to infinity of n - (n + 1) = the sum over n from 1 to infinity of -1

and any infinite sum with a constant term diverges.

Copyright © 1996 Department of Mathematics, Oregon State University

If you have questions or comments, don't hestitate to contact us.