# Series: Polynomial Expansions, Laurent, and Series Expansions

#### Series

In any calculus course, the professor and book will introduce (in a lot of depth) the various types of series, convergence, and tests to determine whether or not a series actually converges. Here is a brief rundown of all of these:

##### What does a Series look like?

$f(x)=\sum _{n = 0} ^\infty$. So, a series is a sequence of partial sums.

##### What is Convergence of a series?

Convergence of a Series means that the series (when everything is added together) comes to one value. For example, $latex \sum _{n = 0} ^{2} n= 0 +1+2=3$. The sum converges. The more interesting sums are infinite sums.

A few Convergence Definitions:

A series $\sum a_n$ is Absolutely Convergent if $$\Sigma |a_n |$$ is convergent.

If $\sum a_n$ is convergent and $\sum |a_n |$ is divergent then $\sum a_n$ is Conditionally Convergent.

##### How do you define a Region of Convergence?

$a-R

##### How do you define a Radius of Convergence

$|x-a|

##### Ways to Test if a Sum Converges

For the following tests we shall use $f_n$ as our sum.

Ratio Test:

$\lim_{n\to\infty} \mid \frac{f_{n+1}}{f_n}\mid =a$

If $a<1$, then $f_n$ is convergent. If $a>1$, $f_n$ is divergent. If $a=1$, then the test fails.

Integral Test:

$\int_{0}^{\infty} f_n dn$

Alternating Series Test:

This is where $\sum a_n$ and $a_n = (-1)^n b_n$ where $b_n \geq 0$ for all n. If $\lim_{n\to\infty}=0$ and ${b_n}\geq {b_{n+1}}$ then the series $\sum a_n$ is convergent.

#### Polynomial Series Expansions

Polynomial series expansions are most accurate around the point they are centered around. For instance, below there is a sine wave that has three polynomial approximations centered around the origin. Notice how each successive approximation improves (each approximation has a different number of terms):