The Magic of Mathematics – Adding up all natural numbers = negative

Came across this and was just amazed, having studied mathematics at university it’s hard to believe that the following statement could be true

\sum\limits_{n=1}^\infty n = - \frac{1}{12}.

However, check out this “proof” to show that it is.

Lets take

S_1 = 1-1+1-1+1-1+1... = \frac{1}{2} .

S_2 = 1-2+3-4+5-6+....

S = 1+2+3+4+5+6....

Now, 2S_2 can be written as

  1-2+3-4+5-6+…
+   1-2+3-4+5-6+…
= 1-1+1-1+1-1+1-…
= S_1.

So we can deduce that 2S_2 = S_1. So it follows that S_2 = \frac{1}{2} S_1=\frac{1}{4}.

Now lets take S - S_2 = 1+2+3+4+5+... - (1-2+3-4+5-...) = 4+8+12+16+... = 4S.

It follows that S - \frac{1}{4} = 4S.

Rearranging gives -\frac{1}{4} = 3S.

Giving S= - \frac{1}{12}.

Now, S = 1+2+3+4+5+....

Therefore,  \sum\limits_{n=1}^\infty n = - \frac{1}{12}. Right?

I found this result absolutely amazing – but this result is used in string theory and quantum mechanics!