In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence...

alternating series may fail the first part of the test. For a generalization, see Dirichlet's test. Leibniz discussed the criterion in his unpublished...

generalized Riemann or Henstock–Kurzweil integral. This can be seen by using Dirichlet's test for improper integrals. It is a good illustration of special techniques...

{2}{|z-1|}}} , hence the assumptions of the Dirichlet's test are fulfilled. Abel's uniform convergence test is a criterion for the uniform convergence...

In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function f {\displaystyle f} to be equal to the sum...

arithmetic progressions Dirichlet's approximation theorem Dirichlet's unit theorem Dirichlet conditions Dirichlet boundary condition Dirichlet's principle Pigeonhole...

out Cauchy's mistake and introduced Dirichlet's test for the convergence of series. It also introduced the Dirichlet function as an example of a function...

The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test. This...

_{0}^{b}\cos \left({\dfrac {t^{3}}{3}}+xt\right)\,dt,} which converges by Dirichlet's test. For any real number x there is a positive real number M such that...

mathematician Peter Gustav Lejeune Dirichlet (1805–1859) is the eponym of many things. Theorems named Dirichlet's theorem: Dirichlet's approximation theorem (diophantine...

{\textstyle \sum _{k=1}^{\infty }2^{k}a_{2^{k}}} converges. Dirichlet's test Abel's test If the series ∑ n = 1 ∞ | a n | {\textstyle \sum _{n=1}^{\infty...

is divergent. The alternating series test can be viewed as a special case of the more general Dirichlet's test: if ( a n ) {\displaystyle (a_{n})} is...

frequently used to prove Abel's theorem and Dirichlet's test. One can also use this technique to prove Abel's test: If ∑ n b n {\textstyle \sum _{n}b_{n}}...

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }a_{n},} where...

The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number...

commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the...

In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes...

Pearson's chi-squared test or Pearson's χ 2 {\displaystyle \chi ^{2}} test is a statistical test applied to sets of categorical data to evaluate how likely...

In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin...

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity lim sup n → ∞...

_{x\to \infty }f(x)} in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral. Final value theorems for...

convergence of the Fourier series holds; introducing Dirichlet's test for the convergence of series; the Dirichlet function as an example that not any function...

test for convergence Cauchy's convergence test Ratio test Direct comparison test Limit comparison test Root test Alternating series test Dirichlet's test...

comparison test A convergence test in which an infinite series or an improper integral is compared to one with known convergence properties. Dirichlet's test Is...

In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series: If lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to...

comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides...

In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing...

The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis. Dirichlet's theorem states that if a and d are coprime natural numbers, then the...

statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated. The term "nonparametric...

In probability theory and statistics, the Dirichlet process (DP) is one of the most popular Bayesian nonparametric models. It was introduced by Thomas...