Is 1+2+3+4+... equal to -1/12? A mind-boggling proof!

[fargoth]

I've see this neat proof:
http://www.youtube.com/watch?v=E-d9...eature=iv&annotation_id=annotation_3085392237 (for some reason the youtube tag didn't work in preview...)
And now I don't see how what I've learned about series convergence is true...
I've been told that if $a_n > b_n \forall n$ then $\sum a_n > \sum b_n$ therefore, if $\sum b_n$ is divergent then, $\sum a_n$ must be too.
Also, If the partial sum diverges, the series is said to be divergent, isn't it?
And what about $a_n \neq 0$ for n that tends to infinity?
So many ways I could show this series diverges, yet he show it's equal to -1/12?

Where am I, or is he, wrong?

 

[DarthMatter]

I think this cannot be true. The sum of all natural numbers up to N equals (as also shown in the end of the video) $N(N+1)/2$. This obviously goes to infinity as N goes to infinity. And of course there is also no way how strictly positive numbers can add up to give a negative result.

 

[fargoth]

That's what I was saying :)
So where is he wrong?

 

[bahamagreen]

A recent thread on this...

 

[Borek]

And as this link was posted we can safely close the thread.