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{{Primary sources|date=September 2010}}

In mathematics, the '''Retkes convergence criterion''', named after Zoltán Retkes, gives [[necessary and sufficient condition]]s for convergence of numerical [[series (mathematics)|series]]. Numerous criteria are known for testing [[convergent series|convergence]]. The most famous of them is the so-called [[Cauchy criterion]], the only one that gives necessary and sufficient conditions.{{Citation needed|date=September 2010}} Under weak restrictions the Retkes criterion gave a new necessary and sufficient condition for the convergence. The criterion will be formulated in the complex settings:

Assume that <math>\quad \{ z_k \}_{k=1}^\infty \subset \bold C </math> and <math>z_i\neq z_j\quad</math> if <math>\quad i\neq j\quad</math>. Then

:<math>\sum_{k=1}^\infty z_k=s \quad\iff\quad \lim_{n\to\infty}\sum_{k=1}^n\frac{z_k^n}{\Pi_k(z_1,\ldots,z_n)}=s</math>

In the above formula <math>\Pi_k(z_1,\ldots,z_n):=(z_k-z_1)(z_k-z_2)\cdots(z_k-z_{k-1})(z_k-z_{k+1})\cdots(z_k-z_n)\quad k=1,\ldots,n.</math>

The equivalence can be proved by using the [[Hermite–Hadamard inequality]].

==References==
* Zoltán Retkes, "An extension of the Hermite–Hadamard [[Inequality (mathematics)|Inequality]]", ''[[Acta Sci. Math. (Szeged)]]'', 74 (2008), pages 95–106.

{{DEFAULTSORT:Retkes Convergence Criterion}}
[[Category:Convergence tests]]

{{Mathanalysis-stub}}

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