## 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):

## 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

## Section 9.5: Analysis

In this final section of Chapter 9 Hungerford discusses the Structure of Finite Groups.  In the previous sections it has been mentioned that groups of order four are isomorphic to  or  and in this section Hungerford connects this idea to larger groups of prime order or  order where p is prime.

## 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 9.3: Analysis

This section is mostly about the Sylow Theorems. While these theorems assist in analyzing nonabelian finite groups Hungerford lacks examples in explaining these theorems. The biggest question then is how do the Sylow Theorems help understand nonabelian finite groups?

## 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 9.1: Analysis

Thus far Direct Products have been mentioned only as a way to use various groups or rings. In this section Hungerford says that the Direct Product of a group (or groups) can tell us much about the group itself. How is this possible? The Direct Product of any number of groups operate only on the section of the product dependent on that group.

## Section 8.5: Analysis

This section on the Simplicity of  doesn't seem to say a lot unless fully understood. The part easiest to understand is that only the group  is not simple. To understand  we must understand what they are.

## Section 8.4: Analysis

Once again the parallels between groups and rings is shown here. Possibly one of the most interesting parts of this section is the First Isomorphism Theorem. This is an idea that has appeared a few times (as rings and every now and then hints of this have appeared in this section on groups).

## 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.