PX155 - F4 - power series and taylor series revisited

- taylor's theorem: if $f (x)$ is a continuous, single-valued function of $x$ with continuous derivatives $f^{'} (x), f^{″} (x) . . . f^{(n)} (x)$ , in a given interval $a \leq x \leq b$ , and if $f^{(n + 1)} (x)$ also exists in this interval, then:

f (x) = f^{'} (a) (x - a) + \frac{1}{2!} f^{″} (a) (x - a)^{2} + . . . + \frac{1}{n!} f^{(n)} (a) (x - a)^{n} + E_{n} (x)

where, $E_{n} (x)$ is a remainder term which describes the error in the approximation of $f (x)$ by the power series of $n + 1$ terms.
- if $lim_{n \to \infty} E_{n} (x) = 0$ , then $f (x)$ can be represented by a power series, that is a taylor series

let's look again at the convergence of a power series. let $y (x)$ be a function that can be described as a power series:

y (x) = b_{0} + b_{1} x + b_{2} x^{n} + . . . + b_{n} x^{n} = \sum_{n = 0}^{\infty} b_{n} x^{n}

use ratio to determine convergence: $k = lim_{n \to \infty} | \frac{b_{n + 1} x^{n + 1}}{b_{n} x^{n}} | = l i m_{n \to \infty} | x | | \frac{b_{n + 1}}{b_{n}} | < 1$ for convergence
$⟹ | x | < lim_{n \to \infty} | b_{n} / b_{n + 1} |$ radius of convergence
or,
$- lim_{n \to \infty} | \frac{b_{n}}{b_{n + 1}} | < x < lim_{n \to \infty} | \frac{b_{n}}{b_{n + 1}} |$