Telescoping series

In mathematics, a telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with either a succeeding or preceding term. Such a technique is also known as the method of differences.

For example, the series

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

simplifies as

:

A pitfall

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

:$0\; =\; sum\_\{n=1\}^infty\; 0\; =\; sum\_\{n=1\}^infty\; (1-1)\; =\; 1\; +\; sum\_\{n=1\}^infty\; (-1\; +\; 1)\; =\; 1,$

is not correct because 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

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

::

* 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, we have

::

:The problem is that the terms do not cancel.

* 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. 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 "x"th "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

:

The sum telescopes, leaving

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

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.