Here's another for you all to ponder... from
http://www.evilmadscientist.com/article.php/SumTrick
Here is a cool math trick that shows that the sum of an infinite number of positive integers is equal to negative one.
Show that the infinite sum S = (1 + 2 + 4 + 8 + 16 + . . .) adds up to S = -1.
Given that S = (1 + 2 + 4 + 8 + 16 + . . .), if you multiply both sides by two, you get
2S = (2+ 4 + 8 + 16 + 32 + . . .).
Then, add one to both sides:
2S + 1 = 1 + (2 + 4 + 8 + 16 + 32 + . . .)
= 1 + 2 + 4 + 8 + 16 + 32 + . . . = S.
Thus, 2S + 1 = S.
To solve for S, subtract 1 from both sides:
2S = S - 1.
Finally, subtract S from both sides:
S = -1.
Isn't just amazing that you can add up so many positive numbers and get a negative answer?
Yes, it's a trick. I found it in the book Mathematical Methods in the Physical Sciences, by Mary L. Boas. Can you figure out why this actually doesn't work?