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Telescoping Sum

A telescoping sum is a sum in which subsequent terms cancel each other, leaving only initial and final terms. For example,

S=sum_(i=1)^(n-1)(a_i-a_(i+1))
(1)
=(a_1-a_2)+(a_2-a_3)+...+(a_(n-2)-a_(n-1))+(a_(n-1)-a_n)
(2)
=(a_1-a_n)
(3)

is a telescoping sum. Creative telescoping constructs such a sum from shifted copies of an addend to obtain a recurrence equation for the original parameter-dependent sum.

See also

Creative Telescoping, Telescoping Series, Zeilberger's Algorithm

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Cite this as:

Weisstein, Eric W. "Telescoping Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TelescopingSum.html

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