# 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?

. 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, . The sum converges. The more interesting sums are infinite sums.

A few Convergence Definitions:

A series  is Absolutely Convergent if  is convergent.

If  is convergent and  is divergent then  is Conditionally Convergent.





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

For the following tests we shall use  as our sum.

Ratio Test:



If , then  is convergent. If ,  is divergent. If , then the test fails.

Integral Test:



Alternating Series Test:

This is where  and  where  for all n. If  and  then the series  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):