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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a telescoping series is a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
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
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
s of
pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...
s) 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 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 Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and work o ...
, ''De dimensione parabolae''.


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 Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
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, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s also admit representation as a difference, which allows telescopic canceling 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 function In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s whose quotient may be broken up into
partial fraction In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction a ...
s, will fail to admit
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
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 In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...
. All of the terms after cancel. * Let ''k,m'' with ''k'' \neq ''m'' be positive integers. Then \sum^\infty_ = \frac \cdot \frac


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 o ...
, a
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
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 and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already ...
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
, and the number of "occurrences" in any time interval having a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
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 In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of the
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''T''''x''. We use the
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
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. Intuitively, if something occurs at least x times before time t, we have to wait at most t for the xth occurrence. 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 \\ &= \frac\times \lim_ \left\lbrack \frac \right\rbrack \\ &= \frac\times \lim_ \left\lbrack \frac + \frac \right\rbrack \\ &=\frac. \end


Other applications

For other applications, see: *
Grandi's series In mathematics, the infinite series , also written : \sum_^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a diverge ...
; *
Proof that the sum of the reciprocals of the primes diverges The sum of the reciprocals of all prime numbers diverges; that is: \sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century ...
, where one of the proofs uses a telescoping sum; *
Fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, a continuous analog of telescoping series; *
Order statistic In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Import ...
, where a telescoping sum occurs in the derivation of a probability density function; *
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
, where a telescoping sum arises in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
; *
Homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
, again in algebraic topology; *
Eilenberg–Mazur swindle In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums. In geometric topology it was introduced by and is often called the Mazur swi ...
, where a telescoping sum of knots occurs; *
Faddeev–LeVerrier algorithm In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial p_A(\lambda)=\det (\lambda I_n - A) of a square matrix, , named after Dmitry Konstantinovic ...
.


References

{{DEFAULTSORT:Telescoping Series Mathematical series