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.

1 Answer
Jan 13, 2017

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# 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.