# Telescoping series

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Telescoping_series
> Markdown URL: https://mediated.wiki/source/Telescoping_series.md
> Source: https://en.wikipedia.org/wiki/Telescoping_series
> Source revision: 1347913760
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), a **telescoping series** is a [series](/source/Series_(mathematics)) whose general term t_n is of the form t_n=a_{n+1}-a_n, i.e. the difference of two consecutive terms of a [sequence](/source/Sequence) (a_n). As a consequence the partial sums of the series only consists of two terms of (a_n) after cancellation.[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](/source/Evangelista_Torricelli), *De dimensione parabolae*.[3]

## Definition

Telescoping [sums](/source/Sum_(mathematics)) are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms.[1][4] Let a_n be the elements of a sequence of numbers. Then

\sum_{n=1}^N \left(a_n - a_{n-1}\right) =  a_N - a_0.

If a_n converges to a limit L, the telescoping [series](/source/Series_(mathematics)) gives:

\sum_{n=1}^\infty \left(a_n - a_{n-1}\right) = L-a_0.

Every series is a telescoping series of its own partial sums.[5]

## Examples

- The product of a finite [geometric series](/source/Geometric_series) with initial term a and common ratio r by the factor (1 - r) yields a telescoping sum:[6]

(1 - r) \sum^n_{k=0} ar^k = \sum^n_{k=0} \left(ar^k - ar^{k+1}\right) = a - a r^{n+1}

When |r| < 1, this allows for a direct calculation of its limit as n \rightarrow \infty and implies:[7]

\sum^\infty_{k=0} ar^k = \frac{a}{1 - r}.

- The series

\sum_{n=1}^\infty\frac{1}{n(n+1)}

is the series of [reciprocals](/source/Multiplicative_inverse) of [pronic numbers](/source/Pronic_number), and it is recognizable as a telescoping series once rewritten in [partial fraction](/source/Partial_fraction_decomposition) form[1] \begin{align} \sum_{n=1}^\infty \frac{1}{n(n+1)} & {} = \sum_{n=1}^\infty \left( \frac{1}{n} - \frac{1}{n+1} \right) \\ {} & {} = \lim_{N\to\infty} \sum_{n=1}^N \left( \frac{1}{n} - \frac{1}{n+1} \right) \\ {} & {} = \lim_{N\to\infty} \left\lbrack {\left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{N} - \frac{1}{N+1}\right) } \right\rbrack \\ {} & {} = \lim_{N\to\infty} \left\lbrack { 1 + \left( - \frac{1}{2} + \frac{1}{2}\right) + \left( - \frac{1}{3} + \frac{1}{3}\right) + \cdots + \left( - \frac{1}{N} + \frac{1}{N}\right) - \frac{1}{N+1} } \right\rbrack \\ {} & {} = \lim_{N\to\infty} \left\lbrack { 1 - \frac{1}{N+1} } \right\rbrack = 1. \end{align}

- Let *k* be a positive integer. Then

\sum^\infty_{n=1} {\frac{1}{n(n+k)}} = \frac{H_k}{k}

where *H**k* is the *k*th [harmonic number](/source/Harmonic_number).

- Let *k* and *m* with *k* \neq *m* be positive integers. Then

\sum^\infty_{n=1} {\frac{1}{(n+k)(n+k+1)\dots(n+m-1)(n+m)}} = \frac{1}{m-k} \cdot \frac{k!}{m!}

where ! denotes the [factorial](/source/Factorial) operation.

- Many [trigonometric functions](/source/Trigonometric_function) also admit representation as differences, which may reveal telescopic canceling between the consecutive terms. Using the [angle addition identity](/source/Angle_addition_identity) for a product of sines,

\begin{align}
\sum_{n=1}^N \sin\left(n\right) & {} = \sum_{n=1}^N \frac{1}{2} \csc\left(\frac{1}{2}\right) \left(2\sin\left(\frac{1}{2}\right)\sin\left(n\right)\right) \\
& {} =\frac{1}{2} \csc\left(\frac{1}{2}\right) \sum_{n=1}^N \left(\cos\left(\frac{2n-1}{2}\right) -\cos\left(\frac{2n+1}{2}\right)\right) \\
& {} =\frac{1}{2} \csc\left(\frac{1}{2}\right) \left(\cos\left(\frac{1}{2}\right) -\cos\left(\frac{2N+1}{2}\right)\right),
\end{align}

which does not converge as N \rightarrow \infty.

## Applications

In [probability theory](/source/Probability_theory), a [Poisson process](/source/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](/source/Memorylessness) [exponential distribution](/source/Exponential_distribution), and the number of "occurrences" in any time interval having a [Poisson distribution](/source/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 *x*th "occurrence". We seek the [probability density function](/source/Probability_density_function) of the [random variable](/source/Random_variable) *T**x*. We use the [probability mass function](/source/Probability_mass_function) for the Poisson distribution, which tells us that

- \Pr(X_t = x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!},

where λ is the average number of occurrences in any time interval of length 1. Observe that the event {*X**t* ≥ x} is the same as the event {*T**x* ≤ *t*}, and thus they have the same probability. Intuitively, if something occurs at least x times before time t, we have to wait at most t for the xth occurrence. The density function we seek is therefore

- \begin{align} f(t) & {} = \frac{d}{dt}\Pr(T_x \le t) = \frac{d}{dt}\Pr(X_t \ge x) = \frac{d}{dt}(1 - \Pr(X_t \le x-1)) \\ \\ & {} = \frac{d}{dt}\left( 1 - \sum_{u=0}^{x-1} \Pr(X_t = u)\right) = \frac{d}{dt}\left( 1 - \sum_{u=0}^{x-1} \frac{(\lambda t)^u e^{-\lambda t}}{u!} \right) \\ \\ & {} = \lambda e^{-\lambda t} - e^{-\lambda t} \sum_{u=1}^{x-1} \left( \frac{\lambda^ut^{u-1}}{(u-1)!} - \frac{\lambda^{u+1} t^u}{u!} \right) \end{align}

The sum telescopes, leaving

- f(t) = \frac{\lambda^x t^{x-1} e^{-\lambda t}}{(x-1)!}.

For other applications, see:

- [Proof that the sum of the reciprocals of the primes diverges](/source/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges), where one of the proofs uses a telescoping sum;
- [Fundamental theorem of calculus](/source/Fundamental_theorem_of_calculus), a continuous analog of telescoping series;
- [Order statistic](/source/Order_statistic), where a telescoping sum occurs in the derivation of a probability density function;
- [Lefschetz fixed-point theorem](/source/Lefschetz_fixed-point_theorem), where a telescoping sum arises in [algebraic topology](/source/Algebraic_topology);
- [Homology theory](/source/Homology_theory), concept in algebraic topology;
- [Eilenberg–Mazur swindle](/source/Eilenberg%E2%80%93Mazur_swindle), where a telescoping sum of knots occurs;
- [Faddeev–LeVerrier algorithm](/source/Faddeev%E2%80%93LeVerrier_algorithm).

## Related concepts

A *telescoping product* is a finite [product](/source/Product_(mathematics)) (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.[8][9] It is the finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms. Let a_n be a sequence of numbers. Then,

\prod_{n=1}^N \frac{a_{n-1}}{a_n} =  \frac{a_0}{a_N}.

If a_n converges to 1, the resulting product gives:

\prod_{n=1}^\infty \frac{a_{n-1}}{a_n} = a_0

For example, the infinite product[8]

\prod_{n=2}^{\infty} \left(1-\frac{1}{n^2} \right)

simplifies as

\begin{align}
\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{align}

## References

1. Apostol, Tom (1967 [1961]). *Calculus, Volume 1*. Second ed. John Wiley & Sons. pp. 386–387.

1. Brian S. Thomson and Andrew M. Bruckner, *Elementary Real Analysis, Second Edition*, CreateSpace, 2008, page 85

1. Weil, André (1989). "Prehistory of the zeta-function". *Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21, 1987*. Boston, Massachusetts: Academic Press. pp. 1–9. [doi:10.1016/B978-0-12-067570-8.50009-3](https://doi.org/10.1016/B978-0-12-067570-8.50009-3). MR 993308.

1. Weisstein, Eric W. ["Telescoping Sum"](https://mathworld.wolfram.com/TelescopingSum.html). *MathWorld*. Wolfram.

1. Ablowitz, Mark J. & Fokas, Athanassios S. (2003). *Complex Variables: Introduction and Applications*. 2nd ed. Cambridge University Press. p. 110. ISBN 978-0-521-53429-1.

1. Apostol, Tom (1967 [1961]). *Calculus, Volume 1*. Second ed. John Wiley & Sons. p. 388.

1. Apostol, Tom (1967 [1961]). *Calculus, Volume 1*. Second ed. John Wiley & Sons. p. 388.

1. ["Telescoping Series - Product"](https://brilliant.org/wiki/telescoping-series-product/). *Brilliant Math & Science Wiki*. Brilliant.org. Retrieved 9 February 2020.

1. Bogomolny, Alexander. ["Telescoping Sums, Series and Products"](https://www.cut-the-knot.org/m/Algebra/TelescopingSums.shtml). *Cut the Knot*. Retrieved 9 February 2020.

---
Adapted from the Wikipedia article [Telescoping series](https://en.wikipedia.org/wiki/Telescoping_series) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Telescoping_series?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.