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.
If so, why and how? If not, why and how? Explanations don't need to be too in-depth; a superficial explanation shall suffice.
1 Answer
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 = 1-2x+3x^2-4x^3+...
which converges for
If we put
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
-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.
