The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12 ?

Ramanujan summation

It is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined...

So today we talk about this,

Sounds interesting…

Sum of all natural numbers existing (and non existing as well) converges to -1/12 ?

i.e,

1 + 2 + 3 + ⋯ + ∞ = -1/12 ?

Solution given over internet :-

Let me define a variable A as 1 - 1 + 1 - 1 + 1 …..

A second variable B = 1 - 2 + 3 - 4 + 5 ….

The third and final variable C = 1 + 2 + 3 + 4 +….. which is the variable we’re interested in computing.

In A, adding values will have the whole sum oscillate between 0 and 1, so we take the average, obtaining A = 1/2.

Now, if we multiply the variable B by two, but shift the numbers over one spot, we see that 2B = 1/2, so B = 1/4.

Then, if we subtract B from C, then we have B - C = 4 + 8 + 12 + …

This can also be written as 4C. So, C - 1/4 = 4C. Multiply by 4 to obtain 4C-1=16C.

Therefore, C = -1/12.

but...

Hence if we discard the proof for this series then obviously the answer to the infinite summation of the natural numbers cannot converge to -1/12 atleast as per the basic mathematics what we understand. Hence, if we go by the definitions of mathematics, it is not possible to achieve that result.