In mathematics, a telescoping series is a series, i.e. the difference of two consecutive terms of a sequence 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, 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., the telescoping series

Every series is a telescoping series of its own partial sums.

Examples

• The product of a finite geometric series yields a telescoping sum: and implies:

  is the series of reciprocalss of pronic numbers, and it is recognizable as a telescoping series once rewritten in partial fraction form

  Let k where Hk is the kth harmonic number.

  Let k and m with k m denotes the factorial operation.

  Many trigonometric functions also admit representation as differences, which may reveal telescopic canceling between the consecutive terms. Using the angle addition identity

Applications

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 Xt be the number of "occurrences" before time t, and let Tx be the waiting time until the xth "occurrence". We seek the probability density function of the random variable Tx. 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 {Xt ≥ x} is the same as the event {Tx ≤ tth occurrence. The density function we seek is therefore

The sum telescopes, leaving

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 in algebraic topology;

• Homology theory, concept in algebraic topology;

• Eilenberg–Mazur swindle, 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.

For example, the infinite product

References