Dirichlet's criterion

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

Dirichlet criterion for convergence

criteria

The series

${\displaystyle \sum \limits _{k=0}^{\infty }a_{k}b_{k}}$

converges with if is a monotonically decreasing zero sequence and is the sequence of the partial sums ${\displaystyle a_{k}\in \mathbb {R} ,b_{k}\in \mathbb {C} }$ ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$ ${\displaystyle (B_{n})_{n\in \mathbb {N} }}$

${\displaystyle B_{n}=\sum \limits _{k=0}^{n}b_{k}}$

is limited .

proof

The following applies (see partial summation )

${\displaystyle \sum \limits _{k=0}^{n}a_{k}b_{k}=a_{n+1}B_{n}+\sum \limits _{k=0}^{n}B_{k}(a_{k}-a_{k+1})}$.

The first summand converges to zero, since it is bounded by a constant according to the assumption and converges to zero. The second summand even converges absolutely , because for all and with it ${\displaystyle B_{n}}$ ${\displaystyle M}$ ${\displaystyle a_{n}}$ ${\displaystyle a_{k}-a_{k+1}\geq 0}$ ${\displaystyle k}$

${\displaystyle \sum \limits _{k=0}^{n}\left|B_{k}(a_{k}-a_{k+1})\right|\leq \sum \limits _{k=0}^{n}M(a_{k}-a_{k+1})=M(a_{0}-a_{n+1})\rightarrow Ma_{0}}$.

Everything is shown.

Dirichlet criterion for uniform convergence

The series

${\displaystyle \sum \limits _{k=0}^{\infty }a_{k}(x)b_{k}(x)}$

is uniformly convergent in the interval if there the partial sums of the series are uniformly bounded and if there the sequence converges uniformly to zero, for every fixed monotone. ${\displaystyle J}$ ${\displaystyle \textstyle \sum b_{k}(x)}$ ${\displaystyle (a_{k}(x))}$ ${\displaystyle x}$

See also

• Abel's criterion
• Leibniz criterion (special case ) ${\displaystyle b_{k}=(-1)^{k}}$

Individual evidence

1. ↑ Harro Heuser : Textbook of Analysis . Part 1. 17th edition. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0777-9 , IV, sentence 33.14, p. 208/643 p .
2. ↑ Konrad Knopp : Theory and application of the infinite series . 6th edition. Springer, Berlin / Heidelberg 1996, ISBN 3-540-59111-7 , pp. 342 ff ./604 S ( edition 1964 ( memento from January 11, 2013 in the web archive archive.today ))