Series whose partial sums eventually only have a fixed number of terms after cancellation
In mathematics, a telescoping series is a series whose general term is of the form , i.e. the difference of two consecutive terms of a sequence .[1]
As a consequence the partial sums only consists of two terms of after cancellation.[2][3] 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.[4]
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 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 ≤ 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
Similar concepts
Telescoping series
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.[6][7]
For example, the infinite product[6]
simplifies as
Other applications
For other applications, see: