telescoping sum

In mathematics, a telescoping series is a series whose general term $t_n$ can be written as $t_n=a_n-a_$, i.e. the difference of two consecutive terms of a sequence $(a_n)$.

As a consequence the partial sums only consists of two terms of $(a_n)$ after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.

For example, the series

:$\sum_^\infty\frac$

(the series of reciprocals of pronic numbers) simplifies as

:$\begin \sum_^\infty \frac & = \sum_^\infty \left( \frac - \frac \right) \\ & = \lim_ \sum_^N \left( \frac - \frac \right) \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack = 1. \end$

In general

Telescoping sums are finite sums in which pairs of consecutive terms cancel each other, leaving only the initial and final terms.

Let $a_n$ be a sequence of numbers. Then,

:$\sum_^N \left(a_n - a_\right) = a_N - a_0$

If $a_n \rightarrow 0$

:$\sum_^\infty \left(a_n - a_\right) = - a_0$

Telescoping products are finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms.

Let $a_n$ be a sequence of numbers. Then,

:$\prod_^N \frac = \frac$

If $a_n \rightarrow 1$

:$\prod_^\infty \frac = a_0$

More examples

* Many trigonometric functions also admit representation as a difference, which allows telescopic cancelling between the consecutive terms.

::$\begin \sum_^N \sin\left(n\right) & = \sum_^N \frac \csc\left(\frac\right) \left(2\sin\left(\frac\right)\sin\left(n\right)\right) \\ & =\frac \csc\left(\frac\right) \sum_^N \left(\cos\left(\frac\right) -\cos\left(\frac\right)\right) \\ & =\frac \csc\left(\frac\right) \left(\cos\left(\frac\right) -\cos\left(\frac\right)\right). \end$

* Some sums of the form

::$\sum_^N$

:where ''f'' and ''g'' are polynomial functions whose quotient may be broken up into partial fractions, will fail to admit summation by this method. In particular, one has

::$\begin \sum^\infty_\frac = & \sum^\infty_\left(\frac+\frac\right) \\ = & \left(\frac + \frac\right) + \left(\frac + \frac\right) + \left(\frac + \frac\right) + \cdots \\ & \cdots + \left(\frac + \frac\right) + \left(\frac + \frac\right) + \left(\frac + \frac\right) + \cdots \\ = & \infty. \end$

:The problem is that the terms do not cancel.

* Let ''k'' be a positive integer. Then

::$\sum^\infty_ = \frac$

:where ''H''_''k'' is the ''k''th harmonic number. All of the terms after 1/(''k'' − 1) cancel.

An application in probability theory

In probability theory, a

Poisson process

is a stochastic process of which the simplest case involves "occurrences" at random times, the waiting time until the next occurrence having a memoryless exponential distribution, and the number of "occurrences" in any time interval having a Poisson distribution whose expected value is proportional to the length of the time interval. Let ''X''_''t'' be the number of "occurrences" before time ''t'', and let ''T''_''x'' be the waiting time until the ''x''th "occurrence". We seek the probability density function of the random variable ''T''_''x''. We use the probability mass function for the Poisson distribution, which tells us that

: $\Pr(X_t = x) = \frac,$

where λ is the average number of occurrences in any time interval of length 1. Observe that the event is the same as the event , and thus they have the same probability. The density function we seek is therefore

: $\begin f(t) & = \frac\Pr(T_x \le t) = \frac\Pr(X_t \ge x) = \frac(1 - \Pr(X_t \le x-1)) \\ \\ & = \frac\left( 1 - \sum_^ \Pr(X_t = u)\right) = \frac\left( 1 - \sum_^ \frac \right) \\ \\ & = \lambda e^ - e^ \sum_^ \left( \frac - \frac \right) \end$

The sum telescopes, leaving

: $f(t) = \frac.$

Similar concepts

Telescoping product

A telescoping product is a finite product (or the partial product of an infinite product) that can be cancelled by method of quotients to be eventually only a finite number of factors.

For example, the infinite product

:$\prod_^ \left(1-\frac \right)$

simplifies as

:$\begin \prod_^ \left(1-\frac \right) &=\prod_^\frac \\ &=\lim_ \prod_^\frac \times \prod_^\frac \\ &= \lim_ \left\lbrack \right\rbrack \times \left\lbrack \right\rbrack \\ &= \lim_ \left\lbrack \frac \right\rbrack \times \left\lbrack \frac \right\rbrack \\ &= \lim_ \left\lbrack \frac \right\rbrack \\ &=\frac. \end$

Other applications

For other applications, see:

* Grandi's series;
* Proof that the sum of the reciprocals of the primes diverges, where one of the proofs uses a telescoping sum;
* Order statistic, where a telescoping sum occurs in the derivation of a probability density function;
* Lefschetz fixed-point theorem, where a telescoping sum arises in algebraic topology;
* Homology theory, again in algebraic topology;
* Eilenberg–Mazur swindle, where a telescoping sum of knots occurs;
*Faddeev–LeVerrier algorithm;
* Fundamental theorem of calculus, a continuous analog of telescoping series.

Notes and references

{{DEFAULTSORT:Telescoping Series
Mathematical series