# Telescoping series

In mathematics, a **telescoping series** is a series whose partial sums eventually only have a fixed number of terms 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

## In general

Let be a sequence of numbers. Then,

and, if

## More examples

- Many trigonometric functions also admit representation as a difference, which allows telescopic cancelling between the consecutive terms.

- Some sums of the form

- where
*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

- The problem is that the terms do not cancel.

- Let
*k*be a positive integer. Then

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

## An application in probability theory

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. The density function we seek is therefore

The sum telescopes, leaving

## 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.
- Fundamental theorem of calculus, a continuous analog of telescoping series.

## Notes and references

- ↑ 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