Taylor Series

What

• A Power Series that converges to f(x)
• Gives us a way to approximate “ugly functions” (such as $e^x$ or trig) with “prettier” ones (polynomials)

Definition

$$T(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(c)(x-c{)}^n}{n!}$$

If $c=0$ then it is called a Maclaurin polynomial

$$M(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(0)x^n}{n!}$$

Taylor Polynomial

When solving, instead of solving an infinite sum a finite amount of terms are used. That is called a Taylor polynomial and is in the form:

$$T_n(x)=f(x)+f^{(1)}(c)+\frac{f^{(2)}(c)(x-c{)}^2}{2!}+\dots +\frac{f^{(n)}(c)(x-c{)}^n}{n!}$$

Lagrange Error Bound

What

• There is a difference between a finite Taylor polynomial and the actual function
  • $R(x)=|T_n(x)-f(x)|$
• This difference is called the Lagrange error bound

Calculation

$$R_n(x)\le E_{max}=\left|\frac{f^{(n+1)}(z)(x-c{)}^{n+1}}{(n+1)!}\right|$$

• n = degree Taylor polynomial
• c = centered at
• x = value attempting to approximate
• z = mystery number where $c\le z\le x$, chosen so $f^{(n+1)}(z)$ is as big as possible
  • $f^{(n+1)}$ is just increasing → let z = x
  • $f^{(n+1)}$ is just decreasing → let z = c