Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed 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

(the series of reciprocals of pronic numbers) simplifies as

Let ${\displaystyle a_{n}}$ be a sequence of numbers. Then,

and, if ${\displaystyle a_{n}\rightarrow 0}$

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 xth "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

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