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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.
In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series: If lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} or if the limit does not exist, then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} diverges.
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 ]
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.
Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in R n can converge. A typical conditionally convergent integral is that on the non-negative real axis of (see Fresnel integral).
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
In the calculus of communicating sequential processes (CSP), divergence occurs when a process performs an endless series of hidden actions. [4] For example, consider the following process, defined by CSP notation: = The traces of this process are defined as: = { , , , , …} = {} Now, consider the following process, which hides the tick event of the Clock process: = As cannot do anything ...