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In mathematics, a **telescoping series** is a series whose general term can be written as , i.e. the difference of two consecutive terms of a sequence .^{[citation needed]}

As a consequence the partial sums only consists of two terms of after cancellation.^{[1]}^{[2]} 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

(the series of reciprocals of pronic numbers) simplifies as

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, *De dimensione parabolae*.^{[3]}

Telescoping sums are finite sums in which pairs of consecutive terms cancel each other, leaving only the initial and final terms.^{[4]}

Let be a sequence of numbers. Then,

If

Telescoping products are finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms.

Let be a sequence of numbers. Then,

If

- Many trigonometric functions also admit representation as a difference, which allows telescopic canceling between the consecutive terms.
- Some sums of the form
*f*and*g*are polynomial functions whose quotient may be broken up into partial fractions, will fail to admit summation by this method. In particular, one has - Let
*k*be a positive integer. Then*H*_{k}is the*k*th harmonic number. All of the terms after 1/(*k*− 1) cancel. - Let
*k,m*with*k**m*be positive integers. Then

In probability theory, a 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 exponential distribution, and the number of "occurrences" in any time interval having a Poisson distribution 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 of the random variable *T*_{x}. We use the probability mass function for the Poisson distribution, which tells us that

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 times before time , we have to wait at most for the occurrence. The density function we seek is therefore

The sum telescopes, leaving

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.^{[5]}^{[6]}

For example, the infinite product^{[5]}

simplifies as

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;
- 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 in algebraic topology;
- Homology theory, again in algebraic topology;
- Eilenberg–Mazur swindle, where a telescoping sum of knots occurs;
- Faddeev–LeVerrier algorithm.

**^**Tom M. Apostol,*Calculus, Volume 1,*Blaisdell Publishing Company, 1962, pages 422–3**^**Brian S. Thomson and Andrew M. Bruckner,*Elementary Real Analysis, Second Edition*, CreateSpace, 2008, page 85**^**Weil, André (1989). "Prehistory of the zeta-function". In Aubert, Karl Egil; Bombieri, Enrico; Goldfeld, Dorian (eds.).*Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21, 1987*. Boston, Massachusetts: Academic Press. pp. 1–9. doi:10.1016/B978-0-12-067570-8.50009-3. MR 0993308.**^**Weisstein, Eric W. "Telescoping Sum".*MathWorld*. Wolfram.- ^
^{a}^{b}"Telescoping Series - Product".*Brilliant Math & Science Wiki*. Brilliant.org. Retrieved 9 February 2020. **^**Bogomolny, Alexander. "Telescoping Sums, Series and Products".*Cut the Knot*. Retrieved 9 February 2020.