1+2+3+4+5+... = -1/12?
The Ramanujan Summation seems to be a paradox. After all, how can the sum of all natural numbers be a negative number, that too a fraction? Well there seems to be some irrefutable evidence behind this thing. And then I will tell you why it is wrong.
First, we will utilize Grandi’s Series:
Note: I have written an article on Grandi’s Series and you can view it over here.
Now this itself is quite fascinating. However, we will see a mistake in this quite seemingly correct evidence. For now, lets continue with the story:
Now, we move on to the final and craziest step of this proof:
And there you have it. Proof that the sum of all natural numbers is -1/12. This summation is famously known as the Ramanujan Summation. Now before I show how this is wrong, I want to show you guys a few other crazy answers we can pull out using this:
I’m not sure if this has a name but it’s quite cool. This is, unfortunately, not my own discovery, but I sadly do not remember where I originally saw this. The next example actually just hit my mind recently. I’m not really sure if it has been discovered before, so pardon me if I call this my own discovery:
Well this seems crazier than any of the stuff we saw before! -1=0?? How can this be possible? Now for this, I suppose you might say this may not be true as it depends on the last term, as it will only be true if the last term is negative. However, we are making these conclusions irrespective of the last term. This is one reason why we are making all these conclusions. So, with that assumption, we can say that all of these sums are correct.
Note: These kind of sums actually have a name! They’re called Cesàro sums. You can read more about them over here. It uses a bunch of complicated mathematical terms, but if you really want to see it, go ahead.
Now, it’s time. After seeing all those crazy conclusions, I really doubt you guys want to disprove them. But, it’s my solemn duty to complete this article. So, it is with great sadness that I now state this: whatever we did till now is wrong. “Why?”, you may ask. It is because we cannot assume the sum of all those numbers to even have a value. For example, when we say B=1–2+3–4+5-…, we are assuming that they have a fixed sum as B. Only because we made that assumption, we could make all these crazy discoveries.
So, these series were quite interesting. Have fun showing them to your friends and stumping them with this logic!