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| In [[mathematics]], a '''telescoping series''' is a [[series (mathematics)|series]] whose partial sums eventually only have a fixed number of terms after cancellation.<ref>[[Tom M. Apostol]], ''Calculus, Volume 1,'' Blaisdell Publishing Company, 1962, pages 422–3</ref><ref>Brian S. Thomson and Andrew M. Bruckner, ''Elementary Real Analysis, Second Edition'', CreateSpace, 2008, page 85</ref> Such a technique is also known as the '''method of differences'''.
| | I'm Randall (23) from Queenzieburn, Great Britain. <br>I'm learning Korean literature at a local high school and I'm just about to graduate.<br>I have a part time job in a post office.<br><br>Feel free to visit my webpage; [http://zhenyoukang.com/plus/guestbook.php Fifa 15 Coin Generator] |
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| For example, the series
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| :<math>\sum_{n=1}^\infty\frac{1}{n(n+1)}</math>
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| simplifies as
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| :<math>\begin{align}
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| \sum_{n=1}^\infty \frac{1}{n(n+1)} & {} = \sum_{n=1}^\infty \left( \frac{1}{n} - \frac{1}{n+1} \right) \\
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| & {} = \lim_{N\to\infty} \sum_{n=1}^N \left( \frac{1}{n} - \frac{1}{n+1} \right) \\
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| & {} = \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 \\
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| & {} = \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 = 1.
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| \end{align}</math>
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| ==In general==
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| Let <math>a_n</math> be a sequence of numbers. Then,
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| :<math>\sum_{n=1}^N \left(a_n - a_{n-1}\right) = a_N - a_{0},</math>
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| and, if <math>a_n \rightarrow 0</math>
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| :<math>\sum_{n=1}^\infty \left(a_n - a_{n-1}\right) = - a_{0}.</math>
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| ==A pitfall==
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| Although telescoping can be a useful technique, there are pitfalls to watch out for:
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| :<math>0 = \sum_{n=1}^\infty 0 = \sum_{n=1}^\infty (1-1) = 1 + \sum_{n=1}^\infty (-1 + 1) = 1\,</math>
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| is not correct because this regrouping of terms is invalid unless the individual terms [[Convergent sequence|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:
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| :<math>
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| \begin{align}
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| \sum_{n=1}^N \frac{1}{n(n+1)} & {} = \sum_{n=1}^N \left( \frac{1}{n} - \frac{1}{n+1} \right) \\
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| & {} = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{N} -\frac{1}{N+1}\right) \\
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| & {} = 1 + \left(- \frac{1}{2} + \frac{1}{2}\right)
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| + \left( - \frac{1}{3} + \frac{1}{3}\right) + \cdots
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| + \left(-\frac{1}{N} + \frac{1}{N}\right) - \frac{1}{N+1} \\
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| & {} = 1 - \frac{1}{N+1}\to 1\ \mathrm{as}\ N\to\infty.
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| \end{align}
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| </math>
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| ==More examples==
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| * Many [[trigonometric function]]s also admit representation as a difference, which allows telescopic cancelling between the consecutive terms.
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| ::<math>
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| \begin{align}
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| \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) \\
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| & {} =\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) \\
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| & {} =\frac{1}{2} \csc\left(\frac{1}{2}\right) \left(\cos\left(\frac{1}{2}\right) -\cos\left(\frac{2N+1}{2}\right)\right).
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| \end{align}
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| </math>
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| * Some sums of the form
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| ::<math>\sum_{n=1}^N {f(n) \over g(n)},</math>
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| :where ''f'' and ''g'' are [[polynomial function]]s whose quotient may be broken up into [[partial fraction]]s, will fail to admit [[summation]] by this method. In particular, we have
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| ::<math>
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| \begin{align}
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| \sum^\infty_{n=0}\frac{2n+3}{(n+1)(n+2)} & {} =\sum^\infty_{n=0}\left(\frac{1}{n+1}+\frac{1}{n+2}\right) \\
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| & {} = \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 \\
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| & {} \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 \\
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| & {} =\infty.
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| \end{align}
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| </math>
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| :The problem is that the terms do not cancel.
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| * Let ''k'' be a positive integer. Then
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| ::<math>\sum^\infty_{n=1} {\frac{1}{n(n+k)}} = \frac{H_k}{k} </math>
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| :where ''H''<sub>''k''</sub> is the ''k''th [[harmonic number]]. All of the terms after 1/(''k'' − 1) cancel.
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| == An application in probability theory ==
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| 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 [[memorylessness|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''<sub>''t''</sub> be the number of "occurrences" before time ''t'', and let ''T''<sub>''x''</sub> be the waiting time until the ''x''th "occurrence". We seek the [[probability density function]] of the [[random variable]] ''T''<sub>''x''</sub>. We use the [[probability mass function]] for the Poisson distribution, which tells us that
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| : <math> \Pr(X_t = x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!}, </math> | |
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| where λ is the average number of occurrences in any time interval of length 1. Observe that the event {''X''<sub>''t''</sub> ≥ x} is the same as the event {''T''<sub>''x''</sub> ≤ ''t''}, and thus they have the same probability. The density function we seek is therefore
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| : <math>
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| \begin{align}
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| 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)) \\ \\
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| & {} = \frac{d}{dt}\left( 1 - \sum_{u=0}^{x-1} \Pr(X_t = u)\right)
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| = \frac{d}{dt}\left( 1 - \sum_{u=0}^{x-1} \frac{(\lambda t)^u e^{-\lambda t}}{u!} \right) \\ \\
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| & {} = \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)
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| \end{align}
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| </math>
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| The sum telescopes, leaving
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| : <math> f(t) = \frac{\lambda^x t^{x-1} e^{-\lambda t}}{(x-1)!}. </math>
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| == Other applications ==
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| For other applications, see:
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| * [[Grandi's series]];
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| * [[Proof that the sum of the reciprocals of the primes diverges]], where one of the proofs uses a telescoping sum;
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| * [[Order statistic]], where a telescoping sum occurs in the derivation of a probability density function;
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| * [[Lefschetz fixed-point theorem]], where a telescoping sum arises in [[algebraic topology]];
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| * [[Homology theory]], again in algebraic topology;
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| * [[Eilenberg–Mazur swindle]], where a telescoping sum of knots occurs.
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| == Notes and references ==
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| {{reflist}}
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| {{DEFAULTSORT:Telescoping Series}}
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| [[Category:Mathematical series]]
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I'm Randall (23) from Queenzieburn, Great Britain.
I'm learning Korean literature at a local high school and I'm just about to graduate.
I have a part time job in a post office.
Feel free to visit my webpage; Fifa 15 Coin Generator