Serial whose partial sums eventually only have a fixed number of terms later on cancellation
In mathematics, a telescoping series is a series whose full general term can be written as , i.e. the difference of ii consecutive terms of a sequence .[ citation needed ]
As a consequence the partial sums just consists of two terms of after cancellation.[i] [ii] The counterfoil technique, with part of each term cancelling with part of the next term, is known as the method of differences.
For example, the series
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(the series of reciprocals of pronic numbers) simplifies as
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An early on argument of the formula for the sum or fractional sums of a telescoping serial tin exist found in a 1644 piece of work by Evangelista Torricelli, De dimensione parabolae.[three]
In general [edit]
A telescoping series of powers. Annotation in the summation sign, , the index due north goes from 1 to m. In that location is no relationship betwixt n and m beyond the fact that both are natural numbers.
Telescoping sums are finite sums in which pairs of sequent terms abolish each other, leaving only the initial and last terms.[4]
Let be a sequence of numbers. Then,
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If
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Telescoping products are finite products in which sequent terms cancel denominator with numerator, leaving just the initial and final terms.
Permit be a sequence of numbers. Then,
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If
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More examples [edit]
- Many trigonometric functions too admit representation as a divergence, which allows scope canceling betwixt the consecutive terms.
- Some sums of the form
where f and g are polynomial functions whose quotient may exist broken up into fractional fractions, will fail to admit summation by this method. In detail, one has
The problem is that the terms do not abolish. - Let k be a positive integer. Then
where H k is the kth harmonic number. All of the terms after 1/(yard − one) cancel. - Allow k,m with k 1000 be positive integers. Then
An application in probability theory [edit]
In probability theory, a Poisson process is a stochastic process of which the simplest instance involves "occurrences" at random times, the waiting time until the adjacent occurrence having a memoryless exponential distribution, and the number of "occurrences" in whatsoever time interval having a Poisson distribution whose expected value is proportional to the length of the time interval. Let X t exist the number of "occurrences" before time t, and permit T x be the waiting time until the tenth "occurrence". We seek the probability density office of the random variable T x . Nosotros use the probability mass function for the Poisson distribution, which tells us that
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where λ is the average number of occurrences in any time interval of length 1. Observe that the consequence {X t ≥ x} is the same as the event {T x ≤ t}, and thus they have the same probability. Intuitively, if something occurs at to the lowest degree times before time , we have to wait at most for the occurrence. The density function we seek is therefore
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The sum telescopes, leaving
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Similar concepts [edit]
Telescoping product [edit]
A telescoping production is a finite product (or the partial product of an space production) that tin be cancelled by method of quotients to be somewhen only a finite number of factors.[5] [6]
For example, the space product[five]
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simplifies every bit
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Other applications [edit]
For other applications, see:
- Grandi'due south series;
- Proof that the sum of the reciprocals of the primes diverges, where i of the proofs uses a telescoping sum;
- Central theorem of calculus, a continuous analog of telescoping series;
- 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;
- Faddeev–LeVerrier algorithm.
References [edit]
- ^ Tom M. Apostol, Calculus, Volume i, Blaisdell Publishing Company, 1962, pages 422–3
- ^ Brian S. Thomson and Andrew M. Bruckner, Simple Real Analysis, 2d Edition, CreateSpace, 2008, page 85
- ^ Weil, André (1989). "Prehistory of the zeta-function". In Aubert, Karl Egil; Bombieri, Enrico; Goldfeld, Dorian (eds.). Number Theory, Trace Formulas and Detached Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21, 1987. Boston, Massachusetts: Academic Press. pp. 1–9. doi:x.1016/B978-0-12-067570-8.50009-3. MR 0993308.
- ^ Weisstein, Eric W. "Telescoping Sum". MathWorld. Wolfram.
- ^ a b "Telescoping Series - Product". Bright Math & Science Wiki. Brilliant.org. Retrieved nine February 2020.
- ^ Bogomolny, Alexander. "Telescoping Sums, Serial and Products". Cutting the Knot . Retrieved 9 February 2020.
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