10. Approximation by Taylor Polynomials

Taylor Polynomials

Definition

The nth order Taylor polynomial of $f$ at the point $a$ is

$T_n(x)={\sum }_{k=0}^n\frac{1}{k!}{f}^{(k)}(a)(x-a{)}^k$

The closer we are to $a$ the better the polynomial approximates.

Strategy

Given an nth order initial value problem we can obtain an mth order approximation

$T_m(x)={\sum }_{k=0}^m\frac{y^{(k)}(a)}{k!}(x-a{)}^k$

• We obtain $y^{(k)}(a)$ for $0\le k\le n-1$ from the initial values.
• The values $y^{(k)}(a)$ for $k\ge n$ are found iteratively by taking sufficient derivatives of the full differential equation, then plugging in known values of $y^{(l)}(a)$ and solving for $y^{(k)}$.