What Is Limits and Infinite Series ?

[Lucretius]

What Is "Limits and Infinite Series"?

I signed up for my classes for spring quarter and I had room for another math class. Since I'm dual majoring in math and physics I figured I should take as much math as I could. I had to sign up for Multivariable Calculus I so I could take Electricity and Magnetism; but I also signed up for "Limits and Infinite Series" which is required for my math major. I did a bit of a google search with nothing really satisfactory; and the course description isn't very detailed:

What exactly is "Limits and Infinite Series"? I'm aware it varies from college to college, but I know other college do have this course. What exactly does it cover? I guess I already know what a limit is from calculus; but an infinite series? I would guess that they're some sort of infinitely long expansion of sums that result in a finite number, like pi...

Is the concept of an infinite series very useful in physics? I am interested in knowing where I would be able to apply this knowledge. Will it help with my study of E&M?

 

[d_leet]

It sounds like a more in depth look at infinite series than is usually covered in a typical calculus 2 class that would cover things like taylor, maclaurin and power series. If you've been taking calculus you had better know what a limit is, so I can only suppose that this class takes a slightly deeper look at things that you have seen before.

 

[Lucretius]

We skipped the chapters on series' unfortunately. No time. We went right from calculating the volumes of solids to differential equations (which I enjoy very much, as I did very well on those sections and understood them.) So I don't really know what those series' are. Perhaps I'll Wikipedia.

 

[HallsofIvy]

In other words, this college is offering a separate course on "limits and series" rather than including it in calculus. A "sequence" is just an infinite "list" of numbers, typically, labled in some way such as a_n. Technically a sequence can be any list of numbers but, typically, we are interested in whether or not a sequence "converges" to some specific number as we go out further on the sequence- "as n goes to infinity". For example, the sequence a_n= 1/n clearly gets smaller and smaller as n gets larger. Its "limit" is 0.

A "series" is a sum of an infinite sequence- if it exists. The sum of the sequence 1/n, $\Sigma \frac{1}{n}$ does not exist because the "partial sums", the sums you get adding a finite number of the terms get larger and larger.

$\Sigma \frac{1}{n^2}$ does "converge" but it is not trivial to show that.

You may have seen "geometric series", $\Sigma_{i=0}^\infty a r^n$ before. It is possible to show that the "partial sum", up to i, only, is
$a\frac{1-r^{n+1}}{1- r}$, and, if -1< r< 1, the partial sums have limit $\frac{a}{1- r}$ and, so, that is the sum of the entire series.

 

[Lucretius]

Ah, I guess this seems somewhat similar to the concept of improper integrals, which we could sum from some a to infinity. It sounds like this is what some of those series' would do.

Convergence, divergence, all that stuff I touched on somewhat when looking at improper integrals. Hopefully that will come in handy.

Thanks for the more in-depth explanation.

 

[arildno]

It definitely WILL come in handy!

 

[Lucretius]

If any of you have taken physics as well (which I'm sure almost all of you have), when does this stuff start being useful? I've never had to do anything involving an infinite series in physics: perhaps in E&M? Or maybe I don't have to use it, but it will make my life easier in calculating a few things?

 

[Eighty]

Integrals (not just improper integrals) are limits of series.

Have you come across Taylor series? They are certainly useful. In physics, it's often assumed that functions are analytic (equal to their Taylor series), and the first few terms are taken to approximate them. This is used in a lot of physics derivations. Taylor series (and Fourier series) are also useful for solving differential equations, which is important in physics.

I'd also like to say that the most important thing to understand about calculus/analysis is the concept of a limit (the "epsilon-delta" definition in particular). Once you understand it, calculus gets a lot easier.

 

[Lucretius]

We hardly touched on the whole "epsilon-delta" bit. I think we spent one day on it when dealing with derivatives, and never saw it again after that. We spent more time on it in high school, and I found it to be, by far, the most confusing part of the entire course.

 

[Lucretius]

Well, my friend and I just spoke with the professor for the class (awesome prof, had him for MATH 124.) He higly recommended I NOT take the class as I was taking MATH 224 (Multivariable Calculus), PHYS 123 (Electricity and Magnetism), the lab, and ENG 101 as well. He said this stuff was going to look like things I've never done before — mostly rigorous proofs of the concepts I learned in 124 and 125.

I understand the concept of a proof from my introduction to logic class and felt that would help me with the course.

He wasn't trying to scare me out of taking the class, but he said that, honestly, 80% of the people who take it struggle with the material. I'm pretty sure I'm going to stick with it, or at least try it out and leave if the going gets too tough. Any opinions?