Week 3+ - Taylor series

Flashcards on Taylor series for multivariable calculus and matrices module (prt 1/A) University of York

Taylor series definition

A smooth function f(x) can be represented as an infinite sum (Taylor series) about the point $x=x_0$

$$f\left(x\right)=\sum_{n=0}^{\infin}\frac{f^{\left(n\right)}\left(x_0\right)}{n!}\left(x-x_0\right)^{n}$$

What is the Maclaurin series?

A Taylor series based at $x_0=0$

What is the Maclaurin series of an exponential function e^x or exp(x).

$$\exp\left(x\right)=\sum_{n=0}^{\infin}\frac{x^{n}}{n!}$$

Taylor’s theorem - integral remainder

A function $f\left(x\right)$ with $\left(N+1\right)$ continuous derivatives in the interval $I$, $f\left(x\right)\in C^{N+1}\left(I\right)$ can be written as

f\left(x\right)=\sum_{n=0}^{N}\frac{f^{\left(n\right)}\left(x_0\right)}{n!}\left(x-x_0\right)^{n}+\frac{1}{N!}\int_{x_0}^{x}\left(x-t\right)^{N}f^{\left(N+1\right)}\left(t\right)\differentialD t

What is the formula for Nth order Taylor polynomial?

$$P_{N}\left(x\right)=\sum_{n=0}^{N}\frac{f^{\left(n\right)}\left(x_0\right)}{n!}\left(x-x_0\right)^{n}$$

Formula for integral remainder of Taylor Series

R_{N}\left(x\right)=\frac{1}{N!}\int_{x_0}^{x}\left(x-t\right)^{N}f^{\left(N+1\right)}\left(t\right)\differentialD t

where $R_{N}$ is the Nth term remainder of f(x).

Taylor’s theorem - derivative remainder/Lagrange

A function $f\left(x\right)$ with $\left(N+1\right)$ continuous derivatives in the interval $I$, $f\left(x\right)\in C^{N+1}\left(I\right)$ can be written as

$f\left(x\right)=\sum_{n=0}^{N}\frac{f^{\left(n\right)}\left(x_0\right)}{n!}\left(x-x_0\right)^{n}+\frac{\left(x-x_0\right)^{N+1}}{\left(N+1\right)!}f^{\left(N+1\right)}\left(c\right)$ for some value $x_0\le c\le x$ .

Mean value theorem for integration

\int_{a}^{b}g\left(x\right)h\left(x\right)\differentialD x=g\left(x_0\right)\int_{a}^{b}h\left(x\right)\differentialD x

Convergence of Taylor series

• consider the derivative form of the remainder term when