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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, 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 function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s 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 fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s, will fail to admit
summation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of ...
, a
Poisson process In probability, statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional t ...

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 In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcu ...
exponential distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, and the number of "occurrences" in any time interval having a
Poisson distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
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 and probability density function of a normal distribution In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real number, real-valued random variable. ...
of the
random variable In probability and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventio ...
''T''''x''. We use the
probability mass function In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
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