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.

How do you define a Region of Convergence?

How do you define a Radius of Convergence

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):

Sine Wave Taylor Series Approximation

Laurent Series

 

Series Expansions

 

Examples