Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms 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.

For example, the series

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

(the series of reciprocals of pronic numbers) simplifies as

\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}

In general

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

\sum _{{n=1}}^{N}\left(a_{n}-a_{{n-1}}\right)=a_{N}-a_{{0}},

and, if $a_n \rightarrow 0$

\sum _{{n=1}}^{\infty }\left(a_{n}-a_{{n-1}}\right)=-a_{{0}}.

More examples

Many trigonometric functions also admit representation as a difference, which allows telescopic cancelling between the consecutive terms.

{\begin{aligned}\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{aligned}}

Some sums of the form

\sum _{{n=1}}^{N}{f(n) \over g(n)},

where f and g are polynomial functions whose quotient may be broken up into partial fractions, will fail to admit summation by this method. In particular, one has

{\begin{aligned}\sum _{{n=0}}^{\infty }{\frac {2n+3}{(n+1)(n+2)}}&{}=\sum _{{n=0}}^{\infty }\left({\frac {1}{n+1}}+{\frac {1}{n+2}}\right)\\&{}=\left({\frac {1}{1}}+{\frac {1}{2}}\right)+\left({\frac {1}{2}}+{\frac {1}{3}}\right)+\left({\frac {1}{3}}+{\frac {1}{4}}\right)+\cdots \\&{}\cdots +\left({\frac {1}{n-1}}+{\frac {1}{n}}\right)+\left({\frac {1}{n}}+{\frac {1}{n+1}}\right)+\left({\frac {1}{n+1}}+{\frac {1}{n+2}}\right)+\cdots \\&{}=\infty .\end{aligned}}

The problem is that the terms do not cancel.

Let k be a positive integer. Then

\sum _{{n=1}}^{\infty }{{\frac {1}{n(n+k)}}}={\frac {H_{k}}{k}}

where H_k is the kth harmonic number. All of the terms after 1/(k − 1) cancel.

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

\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. The density function we seek is therefore

{\begin{aligned}f(t)&{}={\frac {d}{dt}}\Pr(T_{x}\leq t)={\frac {d}{dt}}\Pr(X_{t}\geq x)={\frac {d}{dt}}(1-\Pr(X_{t}\leq 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 ^{u}t^{{u-1}}}{(u-1)!}}-{\frac {\lambda ^{{u+1}}t^{u}}{u!}}\right)\end{aligned}}

The sum telescopes, leaving

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

Other applications

For other applications, see:

Grandi's series;
Proof that the sum of the reciprocals of the primes diverges, where one of the proofs uses a telescoping sum;
Order statistic, where a telescoping sum occurs in the derivation of a probability density function;
Lefschetz fixed-point theorem, where a telescoping sum arises in algebraic topology;
Homology theory, again in algebraic topology;
Eilenberg–Mazur swindle, where a telescoping sum of knots occurs.
Fundamental theorem of calculus, a continuous analog of telescoping series.

Notes and references

↑ Tom M. Apostol, Calculus, Volume 1, Blaisdell Publishing Company, 1962, pages 422–3
↑ Brian S. Thomson and Andrew M. Bruckner, Elementary Real Analysis, Second Edition, CreateSpace, 2008, page 85

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