In mathematics, a telescoping series is a series whose general term <math>t_n</math> is of the form <math>t_n=a_{n+1}-a_n</math>, i.e. the difference of two consecutive terms of a sequence <math>(a_n)</math>. As a consequence the partial sums of the series only consists of two terms of <math>(a_n)</math> after cancellation.

The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.

An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, De dimensione parabolae.

Definition

right|thumb|350px|A telescoping series of powers. Note in the [[summation sign, <math display="inline">\sum</math>, the index n goes from 1 to m. There is no relationship between n and m beyond the fact that both are natural numbers.]]

Telescoping sums are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms. Let <math>a_n</math> be the elements of a sequence of numbers. Then

<math display="block"> \sum_{n=1}^N \left(a_n - a_{n-1}\right) = a_N - a_0.</math>

If <math> a_n </math> converges to a limit <math>L</math>, the telescoping series gives:

<math display="block"> \sum_{n=1}^\infty \left(a_n - a_{n-1}\right) = L-a_0. </math>

Every series is a telescoping series of its own partial sums.

Examples

  • The product of a finite geometric series with initial term <math>a</math> and common ratio <math>r</math> by the factor <math>(1 - r)</math> yields a telescoping sum:<math display="block">(1 - r) \sum^n_{k=0} ar^k = \sum^n_{k=0} \left(ar^k - ar^{k+1}\right) = a - a r^{n+1} </math> When <math>|r| < 1</math>, this allows for a direct calculation of its limit as <math> n \rightarrow \infty</math> and implies: <math display="block"> \sum^\infty_{k=0} ar^k = \frac{a}{1 - r}.</math>
  • The series<math display="block">\sum_{n=1}^\infty\frac{1}{n(n+1)}</math>is the series of reciprocals of pronic numbers, and it is recognizable as a telescoping series once rewritten in partial fraction form It is the finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms. Let <math>a_n</math> be a sequence of numbers. Then,

<math display="block"> \prod_{n=1}^N \frac{a_{n-1{a_n} = \frac{a_0}{a_N}.</math>

If <math>a_n </math> converges to 1, the resulting product gives:

<math display="block"> \prod_{n=1}^\infty \frac{a_{n-1{a_n} = a_0</math>

For example, the infinite product