lunes, 20 de agosto de 2012

M2C2A: Episode 4, 1/2+1/4+1/8...

What is the slowest thing you can think of?
A snail?
A bacteria?
Well, imagine that competing versus a human in a race.
Easy, huh? The human wins.
But what if the human decides to run the distance equivalent to half the race (let's exclude the time he needs to get there), and then run the distance equivalent to a fourth of the race? And then an eight?
He loses.
To explain why we're going to need to use infinitesimal calculus...
Yes, you probably have bad problems with calculus, but I am going to take it easy on you.
First, I'll explain sigma (∑ ).
Let's say you have the following equation:
  ∑    (k-1)+k
This means the ranges of our variable, k, vary from 1 to 3, and each time you solve (k-1)+k, k changes by 1 and the value is added to the last one.
So (k-1)+k is first (when k is 1) equal to 1, and when k is 2, our equation is equal 3, which we add to our last value, 1, to get four, and when k is equal to 3, the result of our equation will be equal to 5, which we add to 4, our last value, to get 9 (Hey! I have accidentally discovered another way to say 32!).
So if I say:
"∑ 2-k"
What do I mean?
I mean 1/2+1/4+1/8...!
But other thing I have to explain is that this is INFINITESIMALLY LESS THAN 1!
If you add 1/2 to 1/4, you get 3/4, 1/4 more to 1, but if you only add then 1/8, you have 7/8, and only need 1/8 more to get 1, but if you only add 1/16...
At the end you might find yourself having to add only 1/∞ more, only to add 1/((1/2)x∞)
So in this race the human keeps running forever and ever only to get nowhere...
-The Roaring Thunder

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