telescoping sum
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 i ...
whose general term t_n is of the form t_n=a_-a_n, i.e. the difference of two consecutive terms of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
(a_n). As a consequence the partial sums of the series 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. 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 ( ; ; 15 October 160825 October 1647) was an Italian people, Italian physicist and mathematician, and a student of Benedetto Castelli. He is best known for his invention of the barometer, but is also known for his advances i ...
, ''De dimensione parabolae''.


Definition

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 a_n be the elements of a sequence of numbers. Then \sum_^N \left(a_n - a_\right) = a_N - a_0. If a_n converges to a limit L, the telescoping
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 i ...
gives: \sum_^\infty \left(a_n - a_\right) = L-a_0. Every series is a telescoping series of its own partial sums.


Examples

* The product of a
geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
with initial term a and common ratio r by the factor (1 - r) yields a telescoping sum, which allows for a direct calculation of its limit:(1 - r) \sum^\infty_ ar^n = \sum^\infty_ \left(ar^n - ar^\right) = a when , r, < 1, so when , r, < 1, \sum^\infty_ ar^n = \frac. * The series\sum_^\infty\fracis the series of reciprocals of pronic numbers, and it is recognizable as a telescoping series once rewritten in partial fraction form \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 * 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 ...
. * Let ''k'' and ''m'' with ''k'' \neq ''m'' be positive integers. Then\sum^\infty_ = \frac \cdot \frac where ! denotes the
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
operation. * 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 differences, which may reveal telescopic canceling between the consecutive terms. Using the angle addition identity for a product of sines,\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 which does not converge as N \rightarrow \infty.


Applications

In
probability theory Probability theory or probability calculus 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 ...
, a
Poisson process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points ...
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 probability distributions. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the geometric and exponential d ...
exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
, 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 if these events occur with a known const ...
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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
of the
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
''T''''x''. We use the
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
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. For other applications, see: * Proof that the sum of the reciprocals of the primes diverges, 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 derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
, 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 Ranking (statistics), rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and ...
, 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 name ...
, 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, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
, 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.


Related concepts

A ''telescoping product'' is a finite product (or the partial product of an infinite product) that can be canceled by the method of quotients to be eventually only a finite number of factors. It is the 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 converges to 1, the resulting product gives: \prod_^\infty \frac = a_0 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. \end


References

{{DEFAULTSORT:Telescoping Series Series (mathematics)