User:Egm4313.s12.team13.fsp.r2.7

Problem Statement
Develop the MacLaurin series (Taylor series at t=0) for

$$e^{t} \qquad \sin t \qquad \ cos t\ $$

To develop any MacLaurin series, it is important to follow the following formula:

$$ f(x) = f(a) + f'(a)\frac{(x-a)}{1!}^{1}\ + f(a)\frac{(x-a)}{2!}^{2} +f'(a)\frac{(x-a)}{3!}^{3}\ + \cdots f^{n}\frac{(x-a)}{n!}^{n}$$

This equation can be simplified by

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

When writing the Maclaurin series for $$ e^{t}$$, using equation 7.0, it looks like

$$ 1 + (1)\frac{(x-0)}{1!}^{1} + (1)\frac{(x-0)}{2!}^{2} + (1)\frac{(x-0)}{3!}^{3}$$

which can be simplified to

Solution R2.7a
$$ e^{t} = 1 + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}\cdots\frac{x^n}{n!} $$

Similarly, starting with the function

$$ \sin t\ $$

the MacLaurin series looks like:

$$ 1\frac{(t-0)}{1!}^{1}+ 0 \frac{(t-0)}{2!}^{2} +1\frac{(t-0)}{3!}^{3} + 0\frac{(t-0)}{4!}^{4} $$

which can be simplified into

Solution R2.7b
$$ \sin t = t - \frac{t^3}{3!} + \frac{t^5}{5!} +\cdots - \frac{t^{n-1}}{(t-1)!} + \frac{t^n}{n!} $$

Finally, starting with

$$ \cos t\ $$, the MacLaurin series looks like

$$ 1 - (0)\frac{(x-0)}{1!}^{1} - (1)\frac{(x-0)}{2!}^{2} + (0)\frac{(x-0)}{3!}^{3} + (1)\frac{(x-0)}{4!}^{4}$$

which can be simplified into

Solution R2.7c
$$ \cos t = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} +\cdots - \frac{t^{n-1}}{(t-1)!} + \frac{t^n}{n!} $$