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Inmathematics, atelescoping series is aseries whose general term${\displaystyle t_n}$ is of the form${\displaystyle t_n=a_{n+1}-a_n}$, i.e. the difference of two consecutive terms of asequence${\displaystyle (a_n)}$. As a consequence the partial sums of the series only consists of two terms of${\displaystyle (a_n)}$ after cancellation.^[1][2]

The cancellation technique, with part of each term cancelling with part of the next term, is known as themethod 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 byEvangelista Torricelli,De dimensione parabolae.^[3]

Telescopingsums are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms.^[1][4] Let${\displaystyle a_n}$ be the elements of a sequence of numbers. Then$${\displaystyle \sum _{n=1}^N\left(a_n-a_{n-1}\right)=a_N-a_0.}$$If${\displaystyle a_n}$ converges to a limit${\displaystyle L}$, the telescopingseries gives:$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\left(a_n-a_{n-1}\right)=L-a_0.}$$

Every series is a telescoping series of its own partial sums.^[5]

• The product of ageometric series with initial term${\displaystyle a}$ and common ratio${\displaystyle r}$ by the factor${\displaystyle (1-r)}$ yields a telescoping sum, which allows for a direct calculation of its limit:^[6]$${\displaystyle (1-r)\sum _{n=0}^{\mathrm{\infty }}a{r}^n=\sum _{n=0}^{\mathrm{\infty }}\left(a{r}^n-a{r}^{n+1}\right)=a}$$when so when$${\displaystyle \sum _{n=0}^{\mathrm{\infty }}a{r}^n=\frac{a}{1-r}.}$$

• The series$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n(n+1)}}$$is the series ofreciprocals ofpronic numbers, and it is recognizable as a telescoping series once rewritten inpartial fraction form^[1]

• Letk be a positive integer. Then$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n(n+k)}=\frac{H_k}{k}}$$ whereH_k is thekthharmonic number.

• Letk andm withk${\displaystyle \ne }$m be positive integers. Then$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{(n+k)(n+k+1)\dots (n+m-1)(n+m)}=\frac{1}{m-k}\cdot \frac{k!}{m!}}$$ where${\displaystyle !}$ denotes thefactorial operation.

• Manytrigonometric functions also admit representation as differences, which may reveal telescopic canceling between the consecutive terms. Using theangle addition identity for a product of sines, which does not converge as$N\to \mathrm{\infty }.$

Inprobability theory, aPoisson process is a stochastic process of which the simplest case involves "occurrences" at random times, the waiting time until the next occurrence having amemorylessexponential distribution, and the number of "occurrences" in any time interval having aPoisson distribution whose expected value is proportional to the length of the time interval. LetX_t be the number of "occurrences" before timet, and letT_x be the waiting time until thexth "occurrence". We seek theprobability density function of therandom variableT_x. We use theprobability mass function for the Poisson distribution, which tells us that

${\displaystyle Pr(X_t=x)=\frac{(\lambda t{)}^x{e}^{-\lambda t}}{x!},}$

where λ is the average number of occurrences in any time interval of length 1. Observe that the event {X_t ≥ x} is the same as the event {T_x ≤t}, and thus they have the same probability. Intuitively, if something occurs at least${\displaystyle x}$ times before time${\displaystyle t}$, we have to wait at most${\displaystyle t}$ for the${\displaystyle xth}$ occurrence. The density function we seek is therefore

The sum telescopes, leaving

${\displaystyle f(t)=\frac{{\lambda }^x{t}^{x-1}{e}^{-\lambda t}}{(x-1)!}.}$

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, a continuous analog of telescoping series;
• Order statistic, where a telescoping sum occurs in the derivation of a probability density function;
• Lefschetz fixed-point theorem, where a telescoping sum arises inalgebraic topology;
• Homology theory, concept in algebraic topology;
• Eilenberg–Mazur swindle, where a telescoping sum of knots occurs;
• Faddeev–LeVerrier algorithm.

Atelescoping product is a finiteproduct (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.^[7][8] It is the finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms. Let${\displaystyle a_n}$ be a sequence of numbers. Then,$${\displaystyle \prod _{n=1}^N\frac{a_{n-1}}{a_n}=\frac{a_0}{a_N}.}$$If${\displaystyle a_n}$ converges to 1, the resulting product gives:$${\displaystyle \prod _{n=1}^{\mathrm{\infty }}\frac{a_{n-1}}{a_n}=a_0}$$

For example, the infinite product^[7]$${\displaystyle \prod _{n=2}^{\mathrm{\infty }}\left(1-\frac{1}{n^2}\right)}$$simplifies as

• Series (mathematics)

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