{"id":4363,"date":"2013-12-16T06:05:45","date_gmt":"2013-12-15T21:05:45","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4363"},"modified":"2014-08-01T19:17:43","modified_gmt":"2014-08-01T10:17:43","slug":"taylor-series","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4363","title":{"rendered":"Taylor series"},"content":{"rendered":"
\n

The Taylor series for f(x) about x = a is defined as<\/p>\n

\\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)
\nwhere \\displaystyle R_n = \\frac{f^n(x - n)^n}{n!}, x0<\/sub> between a and x.(b)
\nis 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
\n\\displaystyle \\frac{f(x) -f(a)}{x - a} = f'(x_0), x0<\/sub> between a and x (c)<\/p>\n

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. <\/p>\n

    \n
  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><img src='https:\/\/s0.wp.com\/latex.php?latex&bg=T&fg=000000&s=0' alt='' title='' class='latex' \/>\\displaystyle \\sin x = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots\\ -\\infty < x < \\infty[\/latex]<\/li> <li><img src='https:\/\/s0.wp.com\/latex.php?latex&bg=T&fg=000000&s=0' alt='' title='' class='latex' \/>\\displaystyle \\cos x = 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\frac{x^6}{6!} + \\cdots\\ -\\infty < x < \\infty[\/latex]<\/li> <li><img src='https:\/\/s0.wp.com\/latex.php?latex&bg=T&fg=000000&s=0' alt='' title='' class='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><img src='https:\/\/s0.wp.com\/latex.php?latex&bg=T&fg=000000&s=0' alt='' title='' class='latex' \/>\\displaystyle \\tan^{-1}x = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots\\ -1 \\le x \\le 1<\/li>\n<\/ol>\n

    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. <\/p>\n<\/blockquote>\n