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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 calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms: [1]. If the infinite series converges and for all sufficiently large n (that is, for all > for some fixed value N), then the infinite series also converges.
Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. ... In calculus, it is common to define notation ...
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
Divergent series (2 C, 15 P) F. Fourier series (31 P) G. ... Sequence transformation; Series expansion; Series multisection; Spectrum continuation analysis; Sturm series;
Fix a polynomial sequence (p n).Define a linear operator Q on polynomials in x by = ().. This determines Q on all polynomials. The polynomial sequence p n is a Sheffer sequence if the linear operator Q just defined is shift-equivariant; such a Q is then a delta operator.
In mathematics, a telescoping series is a series whose general term is of the form = +, i.e. the difference of two consecutive terms of a sequence (). As a consequence the partial sums of the series only consists of two terms of ( a n ) {\displaystyle (a_{n})} after cancellation.
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