Sidebar Sidebar
Become a MacRumors Supporter for $50/year with no ads, ability to filter front page stories, and private forums.

So the sum of all natural numbers from 1 to infinity is...

The thread title is wrong. They are not summing the real numbers. They are summing the natural numbers (or positive integers) but NOT the real numbers.
 
That's a great article. I read it yesterday and watched the video...fascinating and completely non-intuitive. Not only does the sum turn out to be small, but it's negative despite all the components being positive. Thanks for sharing it here.
 
Isn't this "Ramanujan summation"?

fc16cb2a7dfe083c2c52433888b6ccb2.png


If so, then this Web page offers some insight into this sort of "brain melting maths"...

I recently received a few links ([1], [2], [3], [4]) which show some 'brain-melting maths' to prove that the infinite series represented by S = 1 + 2 + 3 + 4 + 5 + ... is equal to a small negative fraction (- 1⁄12 to be specific). I am sure that the brain melting preceded the maths rather than the other way around, else this would not have seen the light of day, let alone so many posts.
Click to expand...
 
I think a lot of the problem most of us normal people have is that we always mentally stop the process at some point short of infinity. And so we inevitably, mentally, come up with an unbelievably massive number.

But true infinity (which by definition, we can never reach) yields a quite different result.

Thought experiments like this always serve to remind me that there is a whole world out there, where "common sense" yields results that are strikingly at odds with the actual results.

Always useful to keep in mind.
 
It's been a long time since I've done any math like this, but I feel a bit queasy watching the video. I may have been able to understand 20 years ago, but I can no longer really explain why - manipulating the infinite series in this way just doesn't feel right, if only because the explanation doesn't take into account anything that's happening at infinity.

In fact, you can easily show 0 = 1 by similarly rearranging infinite series.
 
You must log in or register to reply here.
Register on MacRumors! This sidebar will go away, and you'll see fewer ads.
Top Bottom
Back