Say What?

So, I mentioned before about a problem dealing with Buffy and Batman, and yes it's quite epic. The math itself isn't all that involved really ... well for me anyway. It's just the write up will be tedious, as well as the figures required, and not to mention the animations. Did I just say animations? You're damn straight, I'm pulling out all the stops ... eventually. For right now though I'm a bit too busy with other stuff to devote the time required for such a feat, and I'll probably stay this busy for a good long while. No worries though, I have plenty of things to fill the void until then. Like, for instance, explaining all my freaking steps from my last post. Those of you who are great at math and have enough background probably followed what I did ok. Those of you who are new to sigma notation and infinite series might benefit from a little explanation. There's a lot of little tricks in there that may be really useful to you. I recommend the following to those of you in Calculus 2 or a higher level probability class, ODE, PDE or any higher level class that requires nasty sigma notation funkiness (Ring Theory when you talk about polynomials for instance). Anyone else? Eh, just hope my next post has something more geared toward your level. Sorry, that just comes with the territory. Anyway, here we go:

1. First step I multiplied by [z - (z-1)] which we established was equal to 1, so multiplying every term by 1 doesn't change the series at all. This is a great math trick to know, you can always multiply by 1 (no matter how funky it looks) or add by 0 (no matter how funky it looks).
   
2. Here I pulled the 0th term out of the sum, i.e. I pulled out the term where k=0. Thus we have the 0th term and the rest of the series: 1st term on to infinity. Those of you playing with series, or hell just sums, get used to this trick, it's very useful.
3. Here I just distributed that (-1)^k.
4. Here I did 2 things, one of which may be a little wrong because we don't have absolute convergence. I split up the series (the iffy step) and I rewrote the second piece. Notice if k goes from 1 to infinity, then k+1 goes from 2 to infinity, so I can change the k+1 to a k as long as I change k so it goes from 2 to infinity. Despite the possible iffy step, if this isn't allowed it's ok cause the work doesn't actually require it, I just did it for aesthetic purposes. To fix it jump straight from step (3) to (6) with the same explanation used in (5). It works perfectly, is shorter and less iffy, ugh why didn't I think of that before?
   
5. Here I composed the -(z-1) into the second series, here's how this works. If that series went from 1 to infinity then its 1st term would be -(z-1). Instead we have -(z-1) + the sum starting from 2, so we can suck -(z-1) into that sum as long as we now say that it's starting from 1. Those of you having to play with sums, this is a very very common trick. You move the first term out, play with it, then move it back in.
6. Now I wrote the two sums as one since they both go from 1 to infinity we can (iffily) do this. Another common trick, spit the sum up, play play play, then cram them back together and see what happens. Usually it's magic that happens.
   
7. Factor out the (-1)^k and combine like terms.
8. This step may be a bit hard to follow, even if you have a lot of practice with this stuff. Basically I'm adding up a bunch of terms from k=1 up to infinity. So 1st term + 2nd term + 3rd term on and on and on. Well this step writes the same thing in a different way, basically I split up the even and odd terms so I'm considering the first term to be (1st term + 2nd term) and the second term to be (3rd term + 4th term) and I'm adding these up forever. Here it's the infinity that let's us do this. Plug in a few k's to see that's what is actually going on, and convince yourself that this and the series in step (7) are the same. You'll see this trick a lot in PDE, a lot.
9. The reason I did all that in step (8) is because -1 to an odd power is -1, while -1 to an even power is 1. That's this step.
10. Hey, now we have a ton of zeros added together and...
11. Bam we get z.

Now with all this infinity talk I'm sure you're curious, if that series we talked about acts so wild, then how do I know that an infinite series of zeros is zero? Good question, you should never trust me just because I say something is true. Here's why: take our previous series, at the first term we have just 1, after the second term we have
1 - 1 =0
After the third term we'll have
1 - 1 + 1 = 0 + 1 = 1
After the fourth term we'll have
1 - 1 + 1 - 1 = 1 -1 = 0
See the pattern yet? After an odd number of terms we'll have 1, after an odd number of terms we'll have 0. So this sum is switching back and forth from 1 to 0 over and over, therefore we get trouble when the number of terms approach infinity. On the other hand, a sum of zeros is zero, no matter how many terms you stop at, thus letting that number of terms approach infinity, we'll get zero and nothing else. Is that good enough for you? I can prove it rigorously actually, if someone (out of all 3 of my readers) would like to see it I'll do it ... sometime. For now though, good evening.