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