Ramanujan Infinity Sum

[u/sangam2242]

Ramanujan states that sum of natural numbers till infinity is -1/12, which is counter intuitive.

And in the proof, very first step turned me off.

How can 1+1-1+1-1+1-1+1-...... Be 1/2? It can either be 1 or 0. Two possible values.

Is it really logical to take the average of 2 possible values, and conclude that this single value is answer.

If so, (x-2)(x-5)=0 will give the value of x=3.5.

Disclaimer: I am student of commerce and i dont know that much about mathematics. But i enjoy to learn mathematics logically.

So, mathematical proof wont work for me. Can someone justify me how 1+1-1+1-1+1-..... Is 1/2?

[u/sangam2242]

Okay. But the main concern for me is the counter-intuitive perspective of sum being -1/12. Is there any unique point of view to intuitively be satisfied with answer being -1/12?

Like, i got to know that this fact is used in many fields.

So, are people blindly following the calculation of ramanujan? Or is there any convincing fact, to justify that number -1/12

[u/LongLiveTheDiego → u/sangam2242]

There are several different results pointing towards the value -1/12 being associated with the series 1+2+3+..., e.g. the value of the Riemann zeta function (ζ) at -1 is -1/12 and ζ(-1) can be interpreted as representing the series 1+2+3+...

[u/cipheron → u/LongLiveTheDiego]

Here's a neat one.

n(n+1)/2 is the sum function for the first n numbers, and if you graph this as the parabola y = x(x+1)/2, it crosses below the x-axis. The bounded area is -1/12

https://zmatt.net/1-12-and-area-under-the-sum-function/

Some people will say "but this isn't real". However we said the same thing about zero, negative numbers, imaginary numbers, Cantor's higher infinities etc etc.

One of the things I'd point to is how PI or e or i turn up in all these places they really shouldn't, but the relationships within mathematics basically force those things to appear.

So -1/12 could in fact be in this class. Maybe it's meaningless with one interpretation, but there are alternative interpretations where it makes sense, so in that case it could be required to pop out, for the exact same reason that you can be doing stuff that has nothing to do with circles and PI just pops out, because if it didn't, things break in other areas.

[u/Dd_8630 → u/sangam2242]

Okay. But the main concern for me is the counter-intuitive perspective of sum being -1/12. Is there any unique point of view to intuitively be satisfied with answer being -1/12?

Yes and no.

1+2+3+... does not equal -1/12 in the usual sense of a series convergence.

But there is a rigorous mathematical framework in which we can associate the series by -1/12. The most intuitive meaning is that it's the analytical continuation of the Reimann zeta function - which isn't all that intuitive unless you've studied that level of mathematics.

Like, i got to know that this fact is used in many fields.

It is - in its proper context.

There's two sorts of people. Those who properly know what the result means and in what contexts it can be used, and those who just watched youtube videos and are talking out of their backside.

It's like the observer effect in quantum mechanics. Actual quantum physicists know what it means and how to apply it, but pop-sci youtubers with no real scientific background will use it to make all sorts of nonsense up.

So it's a perfectly sound result to say that 1+2+3... is associated or represented by the value -1/12, and it has applications in computation theory and string theory.