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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, $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<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