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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 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]
A series or, redundantly, an infinite series, is an infinite sum. It is often represented as [ 8 ] [ 15 ] [ 16 ] a 0 + a 1 + a 2 + ⋯ or a 1 + a 2 + a 3 + ⋯ , {\displaystyle a_{0}+a_{1}+a_{2}+\cdots \quad {\text{or}}\quad a_{1}+a_{2}+a_{3}+\cdots ,} where the terms a k {\displaystyle a_{k}} are the members of a sequence of numbers ...
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1 / 4 + 1 / 16 + ⋯ are: + + + + = +. This form can be proved by multiplying both sides by 1 − 1 / 4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs.
The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed ∑ n = 0 ∞ ( − 1 ) n a n {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}} or ∑ n = 0 ∞ ( − 1 ) n + 1 a n {\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}} with a n ...
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (,,, …) defines a series S that is denoted = + + + = =. The n th partial sum S n is the sum of the first n terms of the sequence; that is,