Question

Can anyone tell the proof of 1+2+3+4+5+6+7+8+........upto infinity= -1/12 ?

Answer 1

Noone can proove this thesis, because it is false.

Explanation:

The left side of the expression is a sum of an infinite sequence. Only geometrical sequence can have finite sum of all terms, but this sequence is not a geometrical one therfore it is not convergent (i.e. does not have finite sum).

Answer 2

Here's a "proof" from Srinivasa Ramanujan...

Explanation:

The simplest non-rigorous "proof" is due to Srinivasa Ramanujan and goes roughly as follows:

`c = 1+2+3+4+5+6+7+8+...`

`4c=0+4+0+8+0+12+0+16+...`

Subtracting we get:

`-3c = 1-2+3-4+5-6+7-8+...`

Now:

`1/(1+x)^2 = 1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+...`

So putting `x=1` we find:

`-3c = 1-2+3-4+5-6+7-8+... = 1/(1+1)^2 = 1/4`

Then dividing both ends by `-3` we get:

`c = -1/12`

Note that it is not really valid to manipulate divergent infinite series in these ways.

The calculations above are a shadow of the real derivation of the Ramanujan Sum of the series `1+2+3+4+...`, which is more properly presented using the Riemann Zeta function and analytic continuation. The useful thing about the above non-rigorous derivation is that it gives a very rough sketch of the direction of the proper one.

Ramanujan found ways to formally assign finite values to divergent infinite sums. The methods he developed are used in quantum field theories.