Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.3 Polynomial approximation 4.5 Differentiating Taylor polynomials

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

4.4 Taylor’s polynomials

Suppose that $f(x)\,$ is a function defined for $x\,$ near to $a\,$ , and that we can differentiate $f\,$ as often as we please. We wish to approximate $f(x)\,$ near to $x=a\,$ by a polynomial of degree $\leq n\,$ .

Lemma

Define the Taylor polynomial of $f$ about $a$ of degree $n$ to be

p_{n}(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\dots+a_{n}(x-a)^{n}

where $a_{j}=f^{(j)}(a)/j!$ . Then

p_{n}^{(j)}(a)=f^{(j)}(a)\qquad(j=0,1,\dots,n).