Table of Contents

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*}

observing that certain nominators contain $0$ and can be reduced, we obtain the final formula for $y=cos(x)$:

\begin{eqnarray*} y&=&\sum_{n=2k}^{\infty}(-1)^{k}\frac{x^{k}}{{n!}} \text{,} \forall k\in \mathbb{N} \end{eqnarray*}

Convergence Test

Using the alternating series procedure, the series converges because:

$$ \frac{1}{(2k+1)+1} < \frac{1}{2k+1} $$

and

\begin{eqnarray*} \lim_{n\mapsto\infty}\frac{1}{2k+1} &=& 0 \end{eqnarray*}