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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 convergent series ： ウィキペディア英語版 convergent series In mathematics, a series is the sum of the terms of a sequence of numbers. Given a sequence $\backslash left\; \backslash$, the ''n''th partial sum $S\_n$ is the sum of the first ''n'' terms of the sequence, that is, :$S\_n\; =\; \backslash sum\_^n\; a\_k.$ A series is convergent if the sequence of its partial sums $\backslash left\; \backslash$ converges; in other words, it approaches a given number. In more formal language, a series converges if there exists a limit $\backslash ell$ such that for any arbitrarily small positive number $\backslash varepsilon\; >\; 0$, there is a large integer $N$ such that for all $n\; \backslash ge\; \backslash \; N$, :$\backslash left\; |\; S\_n\; -\; \backslash ell\; \backslash right\; \backslash vert\; \backslash le\; \backslash \; \backslash varepsilon.$ Any series that is not convergent is said to be divergent. == Examples of convergent and divergent series == * The reciprocals of the positive integers produce a divergent series (harmonic series): *: $++++++\backslash cdots\; \backslash rightarrow\; \backslash infty.$ * Alternating the signs of the reciprocals of positive integers produces a convergent series: *: $-\; +\; -\; +\; \backslash cdots\; =\; \backslash ln(2)$ * The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"): *: $++++++\backslash cdots\; \backslash rightarrow\; \backslash infty.$ * The reciprocals of triangular numbers produce a convergent series: *: $++++++\backslash cdots\; =\; 2.$ * The reciprocals of factorials produce a convergent series (see e): *: $\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; =\; e.$ * The reciprocals of square numbers produce a convergent series (the Basel problem): *: $++++++\backslash cdots\; =\; .$ * The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"): *: $++++++\backslash cdots\; =\; 2.$ * Alternating the signs of reciprocals of powers of 2 also produces a convergent series: *: $-+-+-+\backslash cdots\; =\; .$ * The reciprocals of Fibonacci numbers produce a convergent series (see ψ): *: $\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; =\; \backslash psi.$ 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「convergent 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.