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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]
Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (a n) (with n running from 1 to infinity understood) the distance between a n and x approaches 0 as n → ∞, denoted
Once such a sequence is found, a similar question can be asked with f(n) taking the role of 1/n, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series. Using the integral test for convergence, one can show (see below) that, for every natural number k, the series
Rinaldo B. Schinazi: From Calculus to Analysis.Springer, 2011, ISBN 9780817682897, pp. 50 Michele Longo and Vincenzo Valori: The Comparison Test: Not Just for Nonnegative Series.
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 its simplest form, it says that a non-decreasing bounded-above sequence of real numbers ... converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums ...
The test is as follows. Let {g n} be a uniformly bounded sequence of real-valued continuous functions on a set E such that g n+1 (x) ≤ g n (x) for all x ∈ E and positive integers n, and let {f n} be a sequence of real-valued functions such that the series Σf n (x) converges uniformly on E. Then Σf n (x)g n (x) converges uniformly on E.
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