# 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!