Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation. Such a technique is also 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

A pitfall

Although telescoping can be a useful technique, there are pitfalls to watch out for:

is not correct because this regrouping of terms is invalid unless the individual terms converge to 0; see Grandi's series. The way to avoid this error is to find the sum of the first N terms first and then take the limit as N approaches infinity:

More examples

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 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