New Comic And Some Maths
comic diverging numbers sum math
New comic (and topic) have been posted over in the comics forum! Check it out!
To address the maths in the new comic, the first sum is
1 + 2 + 4 + 8 + ... = Sum(2^n), where n = 0, 1, 2, 3, ... all the way up to infinity. Now, if we look at the partial sums of this series, we see that the sum of the first term is 1, the first two terms is 3, the first three terms is 7, first four terms is 15, etc. Clearly, the sum gets bigger the more terms you add on. However, let's say that we have
s = 1 + 2 + 4 + 8 + 16 + ...
We can rewrite this by using the distributive property,
s = 1 + 2 (1 + 2 + 4 + 8 + ...)
But, what's inside the parentheses is clearly the sum we had before, so we can rewrite this as
s = 1 + 2s
Subtracting s from both sides, we get
0 = 1 + s
Therefore, s = -1 = 1 + 2 + 4 + 8 + ... So, while the sum does not converge to a number in the traditional sense, it still has some other meaning that says that it is equivalent to -1. Similarly,
1 - 2 + 4 - 8 + ... = Sum((-2)^n) where n = 0, 1, 2, 3, ... all the way up to infinity, or
s = 1 - 2 + 4 - 8 + ...
We can again rewrite this as
s = 1 - 2(1 - 2 + 4 - 8 + ...)
What's inside the parentheses is, again, the sum, so
s = 1 - 2s, or 3s = 1. Therefore, s = 1/3 So, while the partial sums (1, -1, 3, -5, 11, ...) get larger in magnitude (with alternating +/- signs), the sum is still on some higher level equal to 1/3. Neat, huh?
If this doesn't make sense, that's okay, because it is confusing and highly mathematical. What I think is cool, though, is that you can still show these two statements using alternative mathematical methods, meaning that these values are consistent with different techniques. It's about as awesome to me as how classical physics comes out of quantum mechanics when you take the limit of quantum mechanics for a "large" (classical) system.
I love math and science. 8D