Taylor Series

The Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivates at a single point. Its utility could be to graphically represent functions by selecting a sufficient number of terms and thus decreasing the error gradually.

$\sum_{n=0}^{\infty}=\frac{f^{n}(a)}{n!}(x-a)^n$

or, in expanded form:

$f(a) + x\frac{f'(a)}{1!} + x^{2}\frac{f''(a)}{2!} + \cdots$

For the case $a=0$ , the series is a particular case of the Taylor Series, called the Maclaurin series.

Example: Maclaurin for Cosine Function

Find the Maclaurin expansion for $y=cos(x)$ :

$\begin{eqnarray*} f(0)&=&cos(0)=1 & \\ f'(0)&=&-sin(0)=0 & \\ f''(0)&=&-cos(0)=-1 & \\ f'''(0)&=&sin(0)=0 & \\ f''''(0)&=&cos(0)=1 & \\ \end{eqnarray*}$

which is a repeating pattern.

Now we expand the coefficients:

$\begin{eqnarray*} \sum_{n=0}^{\infty}\frac{f^{k}(0)}{k!}(x-0)^k &=& f(a) + x\frac{f'(a)}{1!} + x^2\frac{f''(a)}{2!} + \cdots & \\ &=& 1 + x\frac{0}{1!} - x^{2}\frac{1}{2!} + x^{3}\frac{0}{3!} + x^{4}\frac{1}{4!} + \cdots \end{eqnarray*}$