MATH1231 - Taylor Series

Taylor Series is the name given to the set of series that:
a) consist of infinitely many polynomial terms
b) are equal to a function.

The Taylor polynomial following is termed the nth Taylor polynomial for f about a. This is applicable when f is n-times differentiable at a.

\begin{align} p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 +\frac{f^{(3)}(a)}{3!}(x-a)^3 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n \end{align}

Hence for a = 0:

\begin{align} p_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 +\frac{f^{(3)}(0)}{3!}x^3 + ... + \frac{f^{(n)}(0)}{n!}x^n \end{align}

Taylor Polynomials can be expressed as a summation:

\begin{align} \sum_{k=0}^{n} \text{ }{\frac{f^{(k)}(a)}{k!}(x-1)^k} \end{align}

Remainder Term

If f has n+1 continuous derivatives on an open interval I containing the point a, then for each x in I:

\begin{equation} f(x) = p_n(x) + R_{n+1}(x) \end{equation}

Where $p_n$ is the nth Taylor Polynomial about a, and $R_{n+1}(x)$ is the remainder term written by:

\begin{align} R_{n+1}(x) = \frac{1}{n!} \int_a^x f^{(n+1)}(t)(x-t)^ndt \end{align}

or in the Lagrange formula:

\begin{align} R_{n+1}(x) = \frac{f^{(n+1)}(c)}{(n+1)!)}(x-a)^{n+1} \end{align}

Second Derivative Test

If f is n times differentiable at a and that $f^'(a) = 0$. If

\begin{equation} f^{''}(a) = f({'''}(a) = /// = f^{(k-1)}(a) = 0 \end{equation}

And if $f^{(k)}(a) \neq 0$ then

  1. a is a local minimum point if k is even and $f^{(k)}(a) > 0$
  2. a is a local maximum point if k is even and $f^{(k)}(a) < 0$
  3. a is a horizontal point of inflexion if k is odd