Telescoping series


In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. 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
simplifies as
A similar concept, telescoping product, is a finite product that can be cancelled by method of quotients to be eventually only a finite number of factors.
For example, the infinite product
simplifies as

In general

Telescoping sums are finite sums in which pairs of consecutive terms cancel each other, leaving only the initial and final terms.
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

More examples

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 is the same as the event, and thus they have the same probability. The density function we seek is therefore
The sum telescopes, leaving

Other applications

For other applications, see: