Increasing Infinite Series

Edit: Thinking about this problem later, the reasoning in this entry is, unfortunately, incorrect.  Well, it's valid for this particular case, but not for all problems of this sort so it's not good enough.  I'd write an entry explaining how the problem is actually solved, but it involves a lot of Calculus concepts that are way too difficult to explain on a blog entry.

I had a flash of brilliance driving home from my Calculus class.

We were given an equation to try to work out as extra credit last Wednesday at the end of class, and I wasn't there as I left half-way through.  Today everyone was talking about it, and nobody had worked it out, so I spent half the class vaguely working on it.  And then, driving home, of course, the answer occured to me in a flash of brilliance.  Or I guess I should say it was a moment of clarity where I was temporarily not-stupid.

You see, we are working on infinite series.  I'll try to explain briefly, if you care.  Which you should, because math is cool.

Anyway, an infinite series is basically adding up infinite amounts of numbers.

Here's a simple one: ∑n.  The ∑ is the sum notation, and normally there would be at the bottom the number you start at and at the top there'd be the number you end at.  So in this case, we'll go from n=1 to ∞ (infinity).  So:

So if we add up to infinity, our ultimate answer would be infinity.  But there are some infinite series that actually add up to a finite (i.e., not infinite) number, or rather, so close to that number that it may as well be that number.  An example is this one:

So that would be 1/2 + 1/4 + 1/8+ 1/16 + 1/32 + 1/64 + ... + 1/2^n.  If you put just 1/2 through 1/64 in your calculator, you'll get about 0.984.  Add the next few terms in the series and it just keeps getting closer and closer to 1.  So we would say that the series ∑1/2^n converges to 1.

There are a few things that distinguish series that converge to a real number from those that diverge (go to infinity) and they're pretty logical.  First, that the numbers have to be getting smaller as you go along the series.  That makes sense; you can't add up ever increasing numbers and expect to get anything other than infinity.  Another feature is that as you get really far along the series, the individual numbers you are adding have to eventually get so small that they're effectively zero.  For instance, if you're adding numbers that keep getting smaller, but never get smaller than, say, 1, then you're still going to end up with infinity.

So in my class we've learned various techniques to determine whether a given series goes to infinity or not and Monday we learned one called the root test, which involves a bunch of stuff you care about even less than this long, stupid math post.  And we were given the following extra credit problem:

And we have to find out whether it converges to a real number or diverges to infinity.  And we all puzzled over it and tried to do complicated integrations and did crazy tests and then driving home it occured to me that, duh, this series doesn't even past the first rule; this series has numbers that are getting bigger, not smaller.

Flashback to algebra.  E is a constant number, about 2.7 something, and it is raised to a negative exponent, which means that you make a fraction, and put it in the denominator and make the exponent positive.  Here's a picture because math doesn't mix with plain text:

Now, we know with fractions that the bigger the denominator is, the smaller the total value of the fraction is, right?  1/4 is smaller than 1/2.  And in our series, n is getting bigger and bigger.  So e^n is getting really big, really fast.  So the fraction 1/e^n is getting really small.

So back to the original equation we were working on.  We are subtracting 1/e^n from 1.  So we are subtracting ever smaller numbers from 1.  So therefore, the total value of 1 minus 1/e^n is increasing.  To illustrate in case you're lost (or even reading anymore), let's try some values of n.  Like, 1 and 2.  So if n=1 then we have (1-1/e^1)^1.  Plug that into my calculator and I get about .63.  Then with n=2, we have (1-1/e^2)^2 which is about .75. 

So the numbers are getting ever bigger, thus breaking the very first and most basic condition of infinite series converging to a real number.