Dirichlet's test

The criterion of Dirichlet is a mathematical convergence criterion for series. It belongs to the group of direct criteria.

Dirichlet criterion for convergence

Criterion

The series

To converge when a zero sequence is monotonically decreasing, and the sequence of partial sums

Is limited.

Evidence

It applies (see Partial summation)

The first term converges to zero, as is the premise of bounded by a constant and converges to zero. The second term converges even absolutely, because for all and thus

So that everything is shown.

Dirichlet criterion for uniform convergence

The series

Is uniformly convergent in the interval when there the partial sums of the series are uniformly bounded and if there the sequence converges uniformly to zero, ie, for every fixed monotone.