Here's a short version of a proof (attributed to Euler):

Euler started with this:

(1) 1 + x + x

^{2}+ x

^{3}+ · · · =1/(1 − x)

He differentiated both sides:

(2) 1 + 2x + 3x

^{2}+ · · · =1/(1 − x)

^{2}

He set x = −1 and got this:

(3) 1 − 2 + 3 − 4 + · · · =1/4

Then Euler considered this function (now known as the Riemann Zeta function):

(4) ζ(s) = 1

^{−s}+ 2

^{−s}+ 3

^{−s}+ 4

^{−s}+ · · ·

He multiplied by 2

^{−s}:

(5) 2

^{−s}ζ(s) = 2

^{−s}+ 4

^{−s}+ 6

^{−s}+ 8

^{−s}+ · · ·

Then he subtracted twice the second equation from the first:

(6) (1 − 2·2

^{−s}) ζ(s) = 1

^{−s}− 2

^{−s}+ 3

^{−s}− 4

^{−s}+ ···

and setting s = −1, he got:

(7) −3(1 + 2 + 3 + 4 + · · · ) = 1 − 2 + 3 − 4 + · · ·

Since he already knew the right-hand side equals 1/4, he concluded:

(8) 1 + 2 + 3 + 4 + · · · = −1/12

So what is wrong in this deduction? Take equation (3).

(3) 1/4 = 1 − 2 + 3 − 4 + 5 - 6 + 7 - 8 · · ·

We can regroup it as (take the first two, then the next two, etc.):

(3a) 1/4 = − 1 − 1 − 1 − 1 · · ·

Or we can regroup (3) as (take the first and third, second and fourth, etc.):

(3b) 1/4 = 4 − 6 + 8 − 10 + 12 − 14 · · ·

Now regroup (3b) as:

(3c) 1/4 = − 2 − 2 − 2 − 2· · ·

Comparing (3a) and (3c), we get that:

(9) 1/2 = 1/4 ???

We can conclude that Euler's deduction was plain wrong.

In all the textbooks, you will find that equation (1) is true only if |x| < 1 or -1 < x < 1, so by setting s equal to -1 in equation (3), we have made equation (1) divergent.

Now in all String Theory textbooks, we are told that ζ(-1)= -1/12 can be derived vigorously.

Stay tuned.

**EDIT:**See: Zeta Function is OK