telescoping sum


A telescoping sum is a sum in which cancellation occurs between subsequent terms, allowing the sum to be expressed using only the initial and final terms.

Formally a telescoping sum is or can be rewritten in the form

S=n=αβ(an-an+1)=aα-aβ+1

where an is a sequence.

Example:

Define S(N)=n=1N1n(n+1). Note that by partial fractions of expressions:

1n(n+1)=1n-1n+1

and thus an=1n in this example.

S(N)=n=1N(1n-1n+1)
=(1-12)++(1n-1n+1)+(1n+1-1n+2)++(1N-1N+1)
=1+(-12+12)++(-1n+1+1n+1)+-1N+1
=1-1N+1
Title telescoping sum
Canonical name TelescopingSum
Date of creation 2013-03-22 14:25:18
Last modified on 2013-03-22 14:25:18
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 8
Author cvalente (11260)
Entry type Definition
Classification msc 40A05
Defines telescope