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Basics of Mathematical Physics Physics

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, $$\sum _{n = 0} ^{2} n= 0 +1+2=3$$. 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?

$$a-R<x<x+R$$

How do you define a Radius of Convergence

$$|x-a|<R$$

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):

Sine Wave Taylor Series Approximation

Laurent Series

 

Series Expansions

 

Examples