In

_{''k''} is the ''k''th harmonic number. All of the terms after 1/(''k'' − 1) cancel.

_{''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 _{''x''}. We use the

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
:$\backslash sum\_^\backslash infty\backslash frac$
(the series of reciprocals of pronic numbers) simplifies as
:$\backslash begin\; \backslash sum\_^\backslash infty\; \backslash frac\; \&\; =\; \backslash sum\_^\backslash infty\; \backslash left(\; \backslash frac\; -\; \backslash frac\; \backslash right)\; \backslash \backslash \; \&\; =\; \backslash lim\_\; \backslash sum\_^N\; \backslash left(\; \backslash frac\; -\; \backslash frac\; \backslash right)\; \backslash \backslash \; \&\; =\; \backslash lim\_\; \backslash left\backslash lbrack\; \backslash right\backslash rbrack\; \backslash \backslash \; \&\; =\; \backslash lim\_\; \backslash left\backslash lbrack\; \backslash right\backslash rbrack\; \backslash \backslash \; \&\; =\; \backslash lim\_\; \backslash left\backslash lbrack\; \backslash right\backslash rbrack\; =\; 1.\; \backslash end$
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, :$\backslash sum\_^N\; \backslash left(a\_n\; -\; a\_\backslash right)\; =\; a\_N\; -\; a\_0$ If $a\_n\; \backslash rightarrow\; 0$ :$\backslash sum\_^\backslash infty\; \backslash left(a\_n\; -\; a\_\backslash right)\; =\; -\; a\_0$ 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, :$\backslash prod\_^N\; \backslash frac\; =\; \backslash frac$ If $a\_n\; \backslash rightarrow\; 1$ :$\backslash prod\_^\backslash infty\; \backslash frac\; =\; a\_0$More examples

* Manytrigonometric 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.
::$\backslash begin\; \backslash sum\_^N\; \backslash sin\backslash left(n\backslash right)\; \&\; =\; \backslash sum\_^N\; \backslash frac\; \backslash csc\backslash left(\backslash frac\backslash right)\; \backslash left(2\backslash sin\backslash left(\backslash frac\backslash right)\backslash sin\backslash left(n\backslash right)\backslash right)\; \backslash \backslash \; \&\; =\backslash frac\; \backslash csc\backslash left(\backslash frac\backslash right)\; \backslash sum\_^N\; \backslash left(\backslash cos\backslash left(\backslash frac\backslash right)\; -\backslash cos\backslash left(\backslash frac\backslash right)\backslash right)\; \backslash \backslash \; \&\; =\backslash frac\; \backslash csc\backslash left(\backslash frac\backslash right)\; \backslash left(\backslash cos\backslash left(\backslash frac\backslash right)\; -\backslash cos\backslash left(\backslash frac\backslash right)\backslash right).\; \backslash end$
* Some sums of the form
::$\backslash sum\_^N$
: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
::$\backslash begin\; \backslash sum^\backslash infty\_\backslash frac\; =\; \&\; \backslash sum^\backslash infty\_\backslash left(\backslash frac+\backslash frac\backslash right)\; \backslash \backslash \; =\; \&\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash cdots\; \backslash \backslash \; \&\; \backslash cdots\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash cdots\; \backslash \backslash \; =\; \&\; \backslash infty.\; \backslash end$
:The problem is that the terms do not cancel.
* Let ''k'' be a positive integer. Then
::$\backslash sum^\backslash infty\_\; =\; \backslash frac$
:where ''H''An application in probability theory

Inprobability 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 ...

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''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''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
: $\backslash Pr(X\_t\; =\; x)\; =\; \backslash frac,$
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
: $\backslash begin\; f(t)\; \&\; =\; \backslash frac\backslash Pr(T\_x\; \backslash le\; t)\; =\; \backslash frac\backslash Pr(X\_t\; \backslash ge\; x)\; =\; \backslash frac(1\; -\; \backslash Pr(X\_t\; \backslash le\; x-1))\; \backslash \backslash \; \backslash \backslash \; \&\; =\; \backslash frac\backslash left(\; 1\; -\; \backslash sum\_^\; \backslash Pr(X\_t\; =\; u)\backslash right)\; =\; \backslash frac\backslash left(\; 1\; -\; \backslash sum\_^\; \backslash frac\; \backslash right)\; \backslash \backslash \; \backslash \backslash \; \&\; =\; \backslash lambda\; e^\; -\; e^\; \backslash sum\_^\; \backslash left(\; \backslash frac\; -\; \backslash frac\; \backslash right)\; \backslash end$
The sum telescopes, leaving
: $f(t)\; =\; \backslash frac.$
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 :$\backslash prod\_^\; \backslash left(1-\backslash frac\; \backslash right)$ simplifies as :$\backslash begin\; \backslash prod\_^\; \backslash left(1-\backslash frac\; \backslash right)\; \&=\backslash prod\_^\backslash frac\; \backslash \backslash \; \&=\backslash lim\_\; \backslash prod\_^\backslash frac\; \backslash times\; \backslash prod\_^\backslash frac\; \backslash \backslash \; \&=\; \backslash lim\_\; \backslash left\backslash lbrack\; \backslash right\backslash rbrack\; \backslash times\; \backslash left\backslash lbrack\; \backslash right\backslash rbrack\; \backslash \backslash \; \&=\; \backslash lim\_\; \backslash left\backslash lbrack\; \backslash frac\; \backslash right\backslash rbrack\; \backslash times\; \backslash left\backslash lbrack\; \backslash frac\; \backslash right\backslash rbrack\; \backslash \backslash \; \&=\; \backslash lim\_\; \backslash left\backslash lbrack\; \backslash frac\; \backslash right\backslash rbrack\; \backslash \backslash \; \&=\backslash frac.\; \backslash end$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