Dave

Indeed. If you know that [imath](a_n) \to a, (b_n) \to b[/imath] then it's fairly easy to prove that [imath](a_n + b_n) \to a+b[/imath]. Doesn't work if one of the limits is undefined though.

You can make the latex much nicer by using \ln, \log, \sin and \cos btw. For example: [math] \frac{d}{dx} \log(\ln x) [/math] (click to view).

You could output the slides from PowerPoint in an image format, then bolt it together in something like iMovie or Movie Maker. Seems to be the easiest way of getting what you're after.

I think the implication was probably natural numbers.

Perhaps combinatorics would be a slightly better choice of words

It's obviously not intended to be a useful proof, it's just a neat piece of trickery

Sorry, I should probably have said that I meant each pi to be prime

The proof is a little tricky. If I remember correctly you have to consider partial sums then do some odd re-arrangement to get the limit out. I'll post again tomorrow when I can actually find my proof of it

Why not break down n into: [imath]n=p_1^{e_1} \cdots p_n^{e_n}[/imath]?

Those sound like things that you can pretty much do in Firefox already. Type "about:config" into the address bar and you can change pretty much every option in there. You'll have to know which ones to look for. The popup blocker sounds pretty much like Adblock.

Also, the motivation behind integrating functions is often not to find the anti-derivative, but the useful fact that if you can integrate a function then you can find its exact area under the curve. For example, [math]\int_{0}^{1} x^2 \, dx = \tfrac{1}{3}[/math]. It's a little tricky at first, but I daresay you'll get used to it after a while - much better than counting squares, and a lot faster!

Yes, but he was more interested in how you actually show that using limits

Probably because you're doing things with the series that aren't necessarily allowed. Re-arranging infinite sums is tricky because the alternating series isn't absolutely convergent; i.e. [imath]\sum a_n[/imath] converges but [imath]\sum |a_n| = \sum \frac{1}{n}[/imath] doesn't converge. You can quite happily re-arrange terms in the series to get two completely different answers. I'll post later, but I'm going to have to look up some results first

Indeed: blatent discrimination

Typically we call the "anti-derivative" of a function the integral of a function. You can denote it like this: [math]\int f(x) \, dx[/math] There's no explicit formula for the integral, and in many cases you can't even integrate a function. You might find it a relatively fun exercise to determine a formula for: [math]\int x^n \, dx[/math] Where n is any number not equal to -1 (why?) I'm reticant to give too much away since you've just started differentiating, but you'll probably touch on it soon enough anyway.

Interestingly enough, there's a course running this year called "Algebraic Geometry". So you can have the best of both worlds, if you like

Why do you assume things have to be "more simple?" Just because they don't seem intuitive doesn't mean that they're incorrect.

It's one of the first proofs I've seen using it. Really it's like using a sledgehammer to crack a nut, but it's a short little proof at least

Not quite. If you're using limits (as I hoped you were in the last question), then [imath]\lim_{x\to 0}\frac{1}{x}[/imath] is undefined. You need to take either the left or right limit to get plus or minus infinity respectively.

Guys, I'd like to remind you about the rule I've imposed on this thread. If you're going to post, then you must wait for at least 3 posts after your last one. I'll start deleting ones that don't obey that from now on, because this thread is just prime for posting unnecessary stuff

We're only concerned with whole numbers, so no

Anything that's interesting really. If you've got them, then either e-mail me or send me a PM.

Ah, that'd be the silly US thing again. Over here in the UK, we consider college to be an interim step between high school and university

As some of you might remember, I used to post a few problems every week for people to solve and reply to. This forum was originally used for the purpose, but obviously that's gone to pot a bit I'd quite like to do the same thing again now we have more visitors around. I plan on holding a small competition every 2 weeks. There will be 3 questions at different levels of mathematics: high school, college and university for those interested in some of the higher maths. What I'm really looking for is some help in gathering possible questions. I'll try and do my best to make up questions where appropriate, but sometimes I won't have the time. I'll probably use the forum this time around instead of issueing PDFs, but I'll moderate posts so that everyone can submit their entry. Any suggestions? I'll write a proper rules list eventually