For a good 20 minutes (or so) of the beginning was spent trying to explain Epsilon-Delta. . . cmon goiz this is 1A stuff. . . but I guess I remember how hard it was for me to learn it. . . but still you shoulda looked it up or something when you were in 1A and now we're wasting time in 1C on this. (Khan Academy did it for me)
Totally honest, I felt like way too much of the class was spent asking stupid questions -stupid as in questions that would not exist if everyone just paid attention. People keep asking for clarification on something that was either A) already gone over at least twice or B) something on the projector already or C) something someone already (or just) asked.
But the professor is so hype so I guess that helps.

the infinite geometric series:
We can write this succinctly as

Now factor out a :
Now note that the above expression can be described as a constant being multiplied to an expression involving a constant multiplied to each term. This new way of writing the series can be expressed as the following:

We now wish to generalize. Retaining the parameters for summation but replacing the constants with variables, we let be the first constant and be the constant by which each term in parentheses is multiplied. (r for ratio) Here is the general case:

Let's be even more general. Instead of having a boundless upper bound, let's let the upper bound be constant .

Now what if we look at some terms. . .

Woah that kinda looks familiar . . .
But what if we're dealing with an infinite series? What if the upper bound is infinity?

And now comes some thinking. Since we don't want to be evaluating a limit every time we deal with an infinite geometric series, we're gonna try to get rid of the limit.
We know the series will either converge or diverge. We need to figure out when and how it does which. Well let's see. The limit is based on the power n. The constant a doesn't really affect it, only the ratio r. If r is some really big positive number, then the expression clearly diverges as n gets big. The same goes if r is a really big negative number (far away from zero). What if r is really small? Well if r is a fraction, then r to the power n will go to zero, and the limit won't get infinitely big. So . . .



Note that for the latter case, the limit simplifies to

You are correct if you said
The key to the definition of a rational number is the first 5 letters of the word itself. A rational number is defined as a number that can be expressed as the ratio of two nonzero integers.

Now let's have some fun with these!
Consider the rational number 4.636363636363636363636363636363636363...
We're gonna prove that it's a rational number by finding the two integers of which it is a ratio of. . . without simply knowing from inspection or multiplying by 100!

Note that we can write the number as such:
[crap it's all squished]

The two integers with ratio 4.63636363636363... are and