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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.
A sequence is a function whose domain is a countable, totally ordered set. [2] The domain is usually taken to be the natural numbers , [ 3 ] although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.
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Illustration of the squeeze theorem When a sequence lies between two other converging sequences with the same limit, it also converges to this limit.. In calculus, the squeeze theorem (also known as the sandwich theorem, among other names [a]) is a theorem regarding the limit of a function that is bounded between two other functions.
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
This series is called balanced if a 1... a k + 1 = b 1...b k q. This series is called well poised if a 1 q = a 2 b 1 = ... = a k + 1 b k, and very well poised if in addition a 2 = −a 3 = qa 1 1/2. The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since
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