Taylor series | Improve Society

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.

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

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.