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 [itex]a_n > b_n \forall n[/itex] then [itex] \sum a_n > \sum b_n [/itex] therefore, if [itex] \sum b_n [/itex] is divergent then, [itex] \sum a_n [/itex] must be too. Also, If the partial sum diverges, the series is said to be divergent, isn't it? And what about [itex] a_n \neq 0 [/itex] 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?
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.