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While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.
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.
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.
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.
For any real sequence , the above results on convergence imply that the infinite series ∑ k = 1 ∞ a k {\displaystyle \sum _{k=1}^{\infty }a_{k}} converges if and only if for every ε > 0 {\displaystyle \varepsilon >0} there is a number N , such that m ≥ n ≥ N imply
In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]
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
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.