Question

Does the sum of all natural numbers REALLY equal -1/12, as stated by Ramanujan?

If so, why and how? If not, why and how? Explanations don't need to be too in-depth; a superficial explanation shall suffice.

Answer 1

No

Explanation:

Srinivasa Ramanujan found a method for associating finite values with divergent series - i.e. series which have no finite sum. The method is called "Ramanujan summation".

Expressed very simply, you could write a "proof" like this:

The Maclaurin series expansion of `1/(1+x)^2` is:

`1/(1+x)^2 = 1-2x+3x^2-4x^3+...`

which converges for `abs(x) < 1`.

If we put `x=1` then we find the non-convergent sum:

`1/4 = 1/(1+1)^2`

`color(white)(1/4) = 1-2+3-4+5-6+...`

`color(white)(1/4) = (1+2+3+4+5+color(white)(0)6+...) -`
`color(white)(1/4 =) (color(white)(0+)4color(white)(+)color(white)(0)+8color(white)(+)color(white)(0)+12+...)`

`color(white)(1/4) = (1+2+3+4+5+6+...) -4(1+2+3+4+5+6+...)`

`color(white)(1/4) = (-3)(1+2+3+4+5+6+...)`

Then divide both ends by `-3` to find:

`-1/12 = 1+2+3+4+5+6+...`

This is not really a proof, but a sort of intuitive motivation and picture of an idea which can be formalised into a method.