Telescoping series

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

As a consequence the partial sums of the series only consists of two terms of

[1][2] 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.

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

be the elements of a sequence of numbers.

converges to a limit

, the telescoping series gives:

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

[5] 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

{\displaystyle xth}

The density function we seek is therefore The sum telescopes, leaving For other applications, see: 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.

[7][8] It is the finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms.

be a sequence of numbers.

converges to 1, the resulting product gives:

For example, the infinite product[7]

simplifies as

lim

{\displaystyle {\begin{aligned}\prod _{n=2}^{\infty }\left(1-{\frac {1}{n^{2}}}\right)&=\prod _{n=2}^{\infty }{\frac {(n-1)(n+1)}{n^{2}}}\\&=\lim _{N\to \infty }\prod _{n=2}^{N}{\frac {n-1}{n}}\times \prod _{n=2}^{N}{\frac {n+1}{n}}\\&=\lim _{N\to \infty }\left\lbrack {{\frac {1}{2}}\times {\frac {2}{3}}\times {\frac {3}{4}}\times \cdots \times {\frac {N-1}{N}}}\right\rbrack \times \left\lbrack {{\frac {3}{2}}\times {\frac {4}{3}}\times {\frac {5}{4}}\times \cdots \times {\frac {N}{N-1}}\times {\frac {N+1}{N}}}\right\rbrack \\&=\lim _{N\to \infty }\left\lbrack {\frac {1}{2}}\right\rbrack \times \left\lbrack {\frac {N+1}{N}}\right\rbrack \\&={\frac {1}{2}}\times \lim _{N\to \infty }\left\lbrack {\frac {N+1}{N}}\right\rbrack \\&={\frac {1}{2}}.\end{aligned}}}