Courses
CTMS-MAT-13: Numerical Methods
Taylor Series

Taylor Series

The Taylor series, or Taylor expansion of a function, is defined as

Definition: Taylor Series

For a function f:R↦Rf : \mathbb{R} \mapsto \mathbb{R} which is infinitely differentiable at a point cc, the Taylor series of f(c)f(c) is given by

∑k=0∞f(k)(c)k!(x−c)k. \begin{equation*} \sum\limits_{k=0}^{\infty} \dfrac{ f^{(k)} \left( c \right) }{k!} \left( x - c \right)^{k}. \end{equation*}

This is a power series, which is convergent for some radius.

Theorem: Taylor's Theorem

For a function f∈Cn+1([a,b])f \in C^{n+1}\left([a, b]\right), i.e. ff is (n+1)(n+1)-times continuously differentiable in the interval [a,b][a, b], then for some cc in the interval, the function can be written as

f(x)=∑k=0nf(k)(c)k!(x−c)k+f(n+1)(ξ)(n+1)!(x−c)n+1 \begin{equation*} f\left( x \right) = \sum\limits_{k=0}^{n} \dfrac{f^{(k)} \left(c\right) }{k!} \left( x- c \right)^{k} + \dfrac{f^{(n+1)} \left( \xi \right) }{\left( n + 1 \right)!} \left( x - c \right)^{n+1} \end{equation*}

for some value ξ∈[a,b]\xi \in \left[ a, b \right] where

lim⁡ξ→cf(n+1)(ξ)(n+1)!(x−c)n+1=0. \begin{equation*} \lim\limits_{\xi \rightarrow c} \dfrac{ f^{(n+1)} \left( \xi \right) }{ \left( n + 1 \right)!} \left( x - c \right)^{n+1} = 0. \end{equation*}
Example:

An .ipynb notebook with an example of the Taylor series for sin⁡(x)\sin\left(x\right) can be accessed online [here].

It can be downloaded from [here] as a python file or downloaded as a notebook from [here].

Theorem: Rolle's Theorem

If a real-valued function ff is continuous on a proper closed interval [a,b][a, b], differentiable on the open interval (a,b)(a, b), and has f (a)=f (b){f (a) = f (b)}, then there exists at least one cc in the open interval (a,b)(a, b) such that

f′(c)=0. f^\prime (c) = 0.
Theorem: Mean Value Theorem

The theorem states that if ff is a continuous function on the closed interval [a,b][a ,b] and differentiable on the open interval (a,b)(a, b), then there exists a point c∈(a,b){c \in (a, b)} such that the tangent at cc is parallel to the secant line through the endpoints (a,f(a)){\big(a, f(a) \big)} and (b,f(b)){\big(b, f(b) \big)}, that is,

f′(c)=f(b)−f(a)b−a. f^\prime (c) = \dfrac{f(b) - f(a)}{b - a}.