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

 

Complex Numbers: The Imaginary Number

The Imaginary Number

Numbers that have real and imaginary parts are complex numbers. They can be expressed in the form where i is  where . Real numbers are actually a special case of complex numbers. Take the above expression. If b is zero then there is no imaginary part and the number is real. As with all special cases the original case has properties that are very much different for example, complex conjugates: Let     (where     denotes that z is a complex number) then the complex conjugate of     is

Continue reading "Complex Numbers: The Imaginary Number"

Section 9.4: Analysis

Conjugacy used here in Section 9.4 seems to be at least partially related to normal groups. For normal groups and does not necessarily equal . Here is said to be conjugate to if there exists an such that . There's a distinct similarity that may have some significance, or we may be able to reword Hungerford's original explanation of Normality using Conjugacy.

Section 9.2: Analysis

In this section Hungerford classifies all Finite Abelian Groups. The first section that may cause some confusion is in Theorem 7.9 where he suddenly calls upon k without previously defining it.  It seems that k may be the the number such that  (in the multiplicative notation) or (in the additive notation).

The Fundamental Theorem of Finite Abelian Groups seems to be very important but Hungerford doesn't explain, or provide more than one example detailing, what it means.

Section 8.3: Analysis

Once again the parallels to rings return in this study of groups. Here Hungerford explains Quotient Groups. They have properties that one might expect such a group to have. The coset notation is probably the most difficult part of this section. Properties of groups have thus far been seen to carry into subgroups and so it is not surprising that that happens here as well.