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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 Power series ： ウィキペディア英語版 Power series In mathematics, a power series (in one variable) is an infinite series of the form :$f(x)\; =\; \backslash sum\_^\backslash infty\; a\_n\; \backslash left(\; x-c\; \backslash right)^n\; =\; a\_0\; +\; a\_1\; (x-c)^1$$+\; a\_2\; (x-c)^2\; +\; a\_3\; (x-c)^3\; +\; \backslash cdots$ where ''an'' represents the coefficient of the ''n''th term, ''c'' is a constant, and ''x'' varies around ''c'' (for this reason one sometimes speaks of the series as being ''centered'' at ''c''). This series usually arises as the Taylor series of some known function. In many situations ''c'' is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form :$$ f(x) = \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots. These power series arise primarily in analysis, but also occur in combinatorics (as generating functions, a kind of formal power series) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument ''x'' fixed at . In number theory, the concept of p-adic numbers is also closely related to that of a power series. ==Examples== Any polynomial can be easily expressed as a power series around any center ''c'', although most of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial $f(x)\; =\; x^2\; +\; 2x\; +\; 3$ can be written as a power series around the center $c=0$ as ::$f(x)\; =\; 3\; +\; 2\; x\; +\; 1\; x^2\; +\; 0\; x^3\; +\; 0\; x^4\; +\; \backslash cdots\; \backslash ,$ or around the center $c=1$ as ::$f(x)\; =\; 6\; +\; 4\; (x-1)\; +\; 1(x-1)^2$$+\; 0(x-1)^3\; +\; 0(x-1)^4\; +\; \backslash cdots\; \backslash ,$ or indeed around any other center ''c''. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials. The geometric series formula ::$\backslash frac\; =\; \backslash sum\_^\backslash infty\; x^n\; =\; 1\; +\; x\; +\; x^2\; +\; x^3\; +\; \backslash cdots,$ which is valid for , is one of the most important examples of a power series, as are the exponential function formula ::$e^x\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; 1\; +\; x\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots,$ and the sine formula ::$\backslash sin(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; x\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac+\backslash cdots,$ valid for all real x. These power series are also examples of Taylor series. Negative powers are not permitted in a power series, for instance $1\; +\; x^\; +\; x^\; +\; \backslash cdots$ is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as $x^$ are not permitted (but see Puiseux series). The coefficients $a\_n$ are not allowed to depend on $x$, thus for instance: :$\backslash sin(x)\; x\; +\; \backslash sin(2x)\; x^2\; +\; \backslash sin(3x)\; x^3\; +\; \backslash cdots\; \backslash ,$ is not a power series. ==Radius of convergence== A power series will converge for some values of the variable ''x'' and may diverge for others. All power series ''f''(''x'') in powers of (''x''-''c'') will converge at ''x'' = ''c''. (The correct value ''f''(''c'') = ''a''0 requires interpreting the expression 00 as equal to 1.) If ''c'' is not the only convergent point, then there is always a number ''r'' with 0 < ''r'' ≤ ∞ such that the series converges whenever |''x'' − ''c''| < ''r'' and diverges whenever |''x'' − ''c''| > ''r''. The number ''r'' is called the radius of convergence of the power series; in general it is given as :$r=\backslash liminf\_\; \backslash left|a\_n\backslash right|^\}$ or, equivalently, $r^=\backslash limsup\_\; \backslash left|a\_n\backslash right|^\}$ (this is the Cauchy–Hadamard theorem; see limit superior and limit inferior for an explanation of the notation). A fast way to compute it is :$r^=\backslash lim\_\backslash left|\backslash right|$ if this limit exists. The series converges absolutely for |''x'' − ''c''| < ''r'' and converges uniformly on every compact subset of . That is, the series is absolutely and compactly convergent on the interior of the disc of convergence. For |''x'' − ''c''| = ''r'', we cannot make any general statement on whether the series converges or diverges. However, for the case of real variables, Abel's theorem states that the sum of the series is continuous at ''x'' if the series converges at ''x''. In the case of complex variables, we can only claim continuity along the line segment starting at ''c'' and ending at ''x''. 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「Power 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.