## domingo, 4 de noviembre de 2012

### M2C2A: Episode 15, Thanks for all the π, but without an e there's no pie...

(Mayority of article by Wikipedia)
(Graph from Wolfram|Alpha)
(Series from Wolfram|Alpha)
Many of you might have heard of e, but not very well the definition. Here it is (for starters):
e=(1+1/∞)
No really, there's no typo. Well, in the other hand:
If you have the equation (1+1/n)n, If n is 1, you have (1+1/1)1, which is equal to 2. If n is 2, you get 2.25. If n is 4, you get 2.44140625... But as you go towards ∞, you get a number closer and closer to e (2.71828182845904523536028747135266249775724709369995...), and when you get to ∞, you get (apparently), e.
Another thing, if a gambler plays a game with a probability of winning of n, n times, for a n that goes up to infinity, the probability of losing every bet is 1/e.
And the following gives e too:
∞
Σ  1/k!
k=0
∞
Σ  ((k-1)2)/k!
k=0
∞
Σ  (2k+1)/(2k)!
k=0

1/2(Σ (k+1)/k!)
k=0

Σ   (k2-2k+1)/k!
k=0

(Σ  ((z-1+k)/k!))/z, where z is any real number (or complex)
k=0

3-(Σ (k+1)/(k+3)!)
k=0
∞
Σ  ((3k)^2+1)/(3k)!
k=0
Also, e is equal to:
1/(2+1/(1+1/(2+1/(1+1/(1+1/(4+1/(1+1/(1+1/(6+1/(1+1/(1+1/(8+1/(1+1/(1+1/(10+1/...)))))))))))))))
Get the pattern?
And it is known that (in radians) e(ix)=cos(x)+i(sin(x)) (which is the base for e(i^π)=-1, but that's another story).
And the biggest non-complex value for x(1/x) is at e.
So e isn't just a bunch of equations, but is the base for a lot of important equations.
-The Roaring Thunder