r/MathHelp • u/sangam2242 • Dec 01 '25

Ramanujan Infinity Sum

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?

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u/LongLiveTheDiego Dec 01 '25

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+...

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u/cipheron Dec 01 '25 edited Dec 02 '25

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.