Editing Telescoping series (section)

==Examples==
* The product of a [[geometric series]] with initial term <math>a</math> and common ratio <math>r</math> by the factor <math>(1 - r)</math> yields a telescoping sum, which allows for a direct calculation of its limit:<ref>{{cite book |last1=Apostol |first1=Tom |title=Calculus, Volume 1 |date=1967 |publisher=John Wiley & Sons |edition=Second |page=388 |orig-date=1961}}</ref><math display="block">(1 - r) \sum^\infty_{n=0} ar^n = \sum^\infty_{n=0} \left(ar^n - ar^{n+1}\right) = a </math>when <math>|r| < 1,</math> so when <math>|r| < 1,</math> <math display="block"> \sum^\infty_{n=0} ar^n = \frac{a}{1 - r}.</math>

* The series<math display="block">\sum_{n=1}^\infty\frac{1}{n(n+1)}</math>is the series of [[Multiplicative inverse|reciprocal]]s of [[pronic number]]s, and it is recognizable as a telescoping series once rewritten in [[Partial fraction decomposition|partial fraction]] form<ref name=":0" /> <math>\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}</math>

* Let ''k'' be a positive integer. Then<math display="block">\sum^\infty_{n=1} {\frac{1}{n(n+k)}} = \frac{H_k}{k} </math> where ''H''<sub>''k''</sub> is the ''k''th [[harmonic number]].

* Let ''k'' and ''m'' with ''k'' <math>\neq</math> ''m'' be positive integers. Then<math display="block">\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!} </math> where <math>! </math> denotes the [[factorial]] operation.

* Many [[trigonometric function]]s also admit representation as differences, which may reveal telescopic canceling between the consecutive terms. Using the [[angle addition identity]] for a product of sines,<math display="block">\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}</math> which does not converge as <math display="inline">N \rightarrow \infty.</math>