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Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem .
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 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 ]
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.
The only divergence for probabilities over a finite alphabet that is both an f-divergence and a Bregman divergence is the Kullback–Leibler divergence. [8] The squared Euclidean divergence is a Bregman divergence (corresponding to the function x 2 {\displaystyle x^{2}} ) but not an f -divergence.
If r = 1, the root test is inconclusive, and the series may converge or diverge. The ratio test and the root test are both based on comparison with a geometric series ...
In computer science, a computation is said to diverge if it does not terminate or terminates in an exceptional state. [1]: 377 Otherwise it is said to converge.In domains where computations are expected to be infinite, such as process calculi, a computation is said to diverge if it fails to be productive (i.e. to continue producing an action within a finite amount of time).
Notably, these series provide examples of infinite sums that converge or diverge arbitrarily slowly. For instance, in the case of k = 2 {\displaystyle k=2} and α = 1 {\displaystyle \alpha =1} , the partial sum exceeds 10 only after 10 10 100 {\displaystyle 10^{10^{100}}} (a googolplex ) terms; yet the series diverges nevertheless.