Taylor series and Maclaurin series
Taylor series and Maclaurin series
In this lesson, we will learn that most functions can be expressed as a Taylor Series. These are power series with a special form, and are centred at a point. If it is centred at 0, then it is called a Maclaurin Series. All of these series require the n'th derivative of the function at point a. We will first apply the Taylor Series formula to some functions. You may notice that trying to find a Taylor Series of a polynomial will just give us back the same polynomial, and not a power series. Then we will learn how to manipulate some of the formulas for some harder functions. Lastly, we will find the Taylor series for sine and cosine, which will requires us to recognize some patterns.
Lessons
Notes:
Note * We can represent a function $f(x)$ about $x=a$ as a Taylor Series. A Taylor Series is in the form
$\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(xa)^n$
where $f^{(n)}(a)$ is the $n$'th derivative at $x=a$. If $a=0$, then we call it a Maclaurin Series. A Maclaurin Series is in the form:
$\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}x^n$
Here are some formulas that may be of use:
$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$
$sin(x)=\sum_{n=0}^{\infty}\frac{(1)^nx^{2n+1}}{(2n+1)!}$
$cos(x)=\sum_{n=0}^{\infty}\frac{(1)^nx^{2n}}{(2n)!}$

1.
Maclaurin Series
Find the Taylor or Maclaurin Series of the following functions without using the formulas: 
2.
Using the Formula to Find the Maclaurin Series
Use the formulas to find the Maclaurin Series for the following functions: