RE: Did you know...?
April 22, 2016 at 5:17 am
(This post was last modified: April 22, 2016 at 5:34 am by Alex K.)
(April 22, 2016 at 5:16 am)pocaracas Wrote: ooooOOO math geekiness!
The sum of all infinite positive natural numbers (1 + 2 + 3 + 4 + .... ) is a negative fractional number!! O.o
Silly String theorists are to blame. But this is only true if you mean by "1+2+3+4+5+6+" not ordinary naive summation but interpret it as a limiting case of representations of the Riemann zeta function. The ordinary series obtained by summing "1+2+3+..." is simply divergent.
To be precise: The sum over (n^-z) from n=1 to infinity for constant z gives the famous Riemann zeta function zeta(z).
Sending z to -1 naively turns this series into "1+2+3+4+5+...", and the zeta function analytically continued in the complex plane to z = -1 is zeta(-1)=-1/12, which has spawned the meme that "1+2+3+4+5+ ... = -1/12", even though the n^-z series which we used to define the Riemann function at points away from z=-1 has no validity there. Still, it's quite mind-boggling that zeta(z->-1) doesn't simply diverge.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition