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*}


fuss/mathematics/calculus/series.txt ยท Last modified: 2017/02/22 18:30 (external edit)

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