日: 2013年12月16日

Taylor series

The Taylor series for f(x) about x = a is defined as

\displaystyle f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)(x - a)^2}{2!} + \cdots + \frac{f^{n -1}(a)(x - a)^{n-1}}{(n -1)!} + R_n(a)
where \displaystyle R_n = \frac{f^n(x - n)^n}{n!}, x0 between a and x.(b)
is called the reminder and where it is supposed that f(x) has derivatives of order n at least. The case where n = 1 is often called law of the mean or mean-value theorem and can be written as
\displaystyle \frac{f(x) -f(a)}{x - a} = f'(x_0), x0 between a and x (c)

The infinite series corresponding to (a), also called the formal Taylor series for f(x), will converge in some interval if \lim\limits_{n \rightarrow \infty}R_n = 0 in this interval. Some important Taylor series together with their intervals of convergence are as follows.

  1. \displaystyle e^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +\frac{x^4}{4!} + \cdots\ -\infty < x < \infty[/latex]</li> <li>[latex]\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\ -\infty < x < \infty[/latex]</li> <li>[latex]\displaystyle \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\ -\infty < x < \infty[/latex]</li> <li>[latex]\displaystyle \ln(1 + x) = x - \frac{x^2}{2!} + \frac{x^3}{3!} - \frac{x^4}{4!} + \cdots\ -1 < x \le 1[/latex]</li> <li>[latex]\displaystyle \tan^{-1}x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\ -1 \le x \le 1

A series of the form \sum_{n=0}^{\infty}c_n(x - a)^n is often called a power series. Such power series are uniformly convergent in any interval which lies entirely within the interval of convergence.

テーラー級数

　x = a における f(x) のテーラー級数は下記のように定義されます．

\displaystyle f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)(x - a)^2}{2!} + \cdots + \frac{f^{n -1}(a)(x - a)^{n-1}}{(n -1)!} + R_n(a)
ここで \displaystyle R_n = \frac{f^n(x - n)^n}{n!}, x0 は a と x の範囲内(b)
は reminder と呼ばれ， f(x) が最低でも n 次の微分係数を持つと想定します．n = 1 の場合は特に平均の法則または平均値の定理と呼ばれ，次のように記述します．
\displaystyle \frac{f(x) -f(a)}{x - a} = f'(x_0), x0 は a と x の範囲内(c)

　(a) に対応した無限級数は f(x) の正規テーラー級数とも呼ばれ，必ずある区間に収束します．仮に \lim\limits_{n \rightarrow \infty}R_n = 0 がこの区間にあるなら．いくつかの重要なテーラー級数と収束値をその区間とともに示します．

  1. \displaystyle e^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +\frac{x^4}{4!} + \cdots\ -\infty < x < \infty[/latex]</li> <li>[latex]\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\ -\infty < x < \infty[/latex]</li> <li>[latex]\displaystyle \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\ -\infty < x < \infty[/latex]</li> <li>[latex]\displaystyle \ln(1 + x) = x - \frac{x^2}{2!} + \frac{x^3}{3!} - \frac{x^4}{4!} + \cdots\ -1 < x \le 1[/latex]</li> <li>[latex]\displaystyle \tan^{-1}x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\ -1 \le x \le 1

　ある形の級数 \sum_{n=0}^{\infty}c_n(x - a)^n はしばしばべき級数と呼ばれます．そのようなべき級数はどの区間にも一様収束し，それは収束区間内全体にあります．