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A power series is a series of the form = (). The Taylor series at a point of a function is a power series that, in many cases, converges to the function in a neighborhood of . For example, the series
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.
1 Examples. 2 Convergence. 3 See also. ... In calculus, a function series is a series where each of its terms is a function, not just a real or complex number. Examples
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
The Maclaurin series of the logarithm function (+) is conditionally convergent for x = 1. The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence.
Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for the Fibonacci numbers, see the article on Examples of generating functions. One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting.
The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis . It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series ...