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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Retkes convergence criterion ： ウィキペディア英語版 Retkes convergence criterion In mathematics, the Retkes convergence criterion, named after Zoltán Retkes, gives necessary and sufficient conditions for convergence of numerical series. Numerous criteria are known for testing convergence. The most famous of them is the so-called Cauchy criterion, the only one that gives necessary and sufficient conditions. 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 $\backslash quad\; \backslash \_^\backslash infty\; \backslash subset\; \backslash bold\; C$ and $z\_i\backslash neq\; z\_j\backslash quad$ if $\backslash quad\; i\backslash neq\; j\backslash quad$. Then :$\backslash sum\_^\backslash infty\; z\_k=s\; \backslash quad\backslash iff\backslash quad\; \backslash lim\_\backslash sum\_^n\backslash frac=s$ In the above formula $\backslash Pi\_k(z\_1,\backslash ldots,z\_n):=(z\_k-z\_1)(z\_k-z\_2)\backslash cdots(z\_k-z\_)(z\_k-z\_)\backslash cdots(z\_k-z\_n)\backslash quad\; k=1,\backslash ldots,n.$ The equivalence can be proved by using the Hermite–Hadamard inequality. ==References== * Zoltán Retkes, "An extension of the Hermite–Hadamard Inequality", ''Acta Sci. Math. (Szeged)'', 74 (2008), pages 95–106. * Encyclopedia of Mathematics. "Cauchy Criteria". European Mathematical Society. Retrieved 4 March 2014. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Retkes convergence criterion」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.