Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.5 Differentiating Taylor polynomials 4.7 Maclaurin’s Theorem

Style control - access keys in brackets

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

4.6 Taylor expansions

Taylor’s Theorem

Let $f\,$ be suitably differentiable. Then

f(x)=f(a)+f^{\prime}(a)(x-a)+{{f^{\prime\prime}(a)}\over{2!}}(x-a)^{2}+\dots+{% {f^{(n)}(a)}\over{n!}}(x-a)^{n}+R_{n}(x)

where the remainder term satisfies $R_{n}(x)=f^{(n+1)}(c)(x-a)^{n+1}/(n+1)!\,$ for some $c$ between $a$ and $x$ . If $R_{n}(x)\rightarrow 0$ as $n\rightarrow\infty$ , then

f(x)=f(a)+f^{\prime}(a)(x-a)+{{f^{\prime\prime}(a)}\over{2!}}(x-a)^{2}+\dots+{% {f^{(n)}(a)}\over{n!}}(x-a)^{n}+\dots.