To find , write each term as . Then

This is analogous to integration: .

And practically exactly the same thing as the finite calculus version of integration, summation.

All series are telescoping series!

e.g.

Find the sum of .

To convert this to a telescoping series, we need to find a way of expressing each term as .

Maybe the e.g. term can be extended in both directions, and , and expressed as the difference of multiples of these, i.e. and .

Noting that , we can use the **multiply by 1 cleverly** tool to find the desired expression.

In terms of falling powers, as is usual in finite calculus:

(This is the finite calculus version of the more familiar .)

**Applications etc**

Common ad nauseam in the telescoping literature is the one arising from , which I was thus going to avoid here, until I found an interesting version due to Marc Frantz.

The th telephone pole appears at . From the side ratios of similar triangles we get:

The th gap, between and , is:

The poles appear between and , i.e. the sum of the series (of gaps) is .

Partial sums of a geometric series

.

If , then . If , then

If then as , and the series converges to .

We need an expression with and terms, for some function .

Since , and , we expect to be an expression in , so try that.

**Pascal’s method**(1654) of summing powers of positive integers.

First find , then use to find , to find and so on.

**useful identities**

**more**

Telescoping Sums, Series and Products at cut-the-knot.org

Paul Zeitz – The Art and Craft of Problem Solving

Thomas Osler – Some Long Telescoping Series

J. Marshall Ash and Stefan Catoiu – Telescoping, rational-valued series, and zeta functions, Trans. Amer. Math. Soc. 357 (2005), p3339-58 PDF

Marc Frantz – The Telescoping Series in Perspective, *Mathematics Magazine*, Vol. 71, No. 4, Oct 1998