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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 Alternating series ： ウィキペディア英語版 Alternating series In mathematics, an alternating series is an infinite series of the form :$\backslash sum\_^\backslash infty\; (-1)^n\backslash ,a\_n$ or $\backslash sum\_^\backslash infty\; (-1)^\backslash ,a\_n$ with ''an'' > 0 for all ''n''. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. ==Examples== The geometric series 1/2 %E2%88%92 1/4 %2B 1/8 %E2%88%92 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: :$$ \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, :$\backslash sin\; x\; =\; \backslash sum\_^\backslash infty\; (-1)^n\; \backslash frac$, and :$\backslash cos\; x\; =\; \backslash sum\_^\backslash infty\; (-1)^n\; \backslash frac\; .$ When the alternating factor (–1)n is removed from these series one obtains the hyperbolic functions sinh and cosh used in calculus. For integer or positive index α the Bessel function of the first kind may be defined with the alternating series :$J\_\backslash alpha(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash right)\}^$   where Γ(''z'') is the gamma function. If ''s'' is a complex number, the Dirichlet eta function is formed as an alternating series :$\backslash eta(s)\; =\; \backslash sum\_^\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$ that is used in analytic number theory. 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「Alternating series」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.