And this sum diverges. So how can it be equal to ? Unless there is some secret mathematical trick that I am not aware of, I don't understand what's up with this statement which is by any means wrong.
Any clues ?
Well, that's what I think. I don't care what all those integrals on Wikipedia tell me, for me this sequence is definitely equal to for odd and for even , regardless of the value of n. Even if n goes towards infinity.
No one can provide a proof for something that makes no sense. I don't have such a high level of abstraction, sorry. I'm just going to leave this one unsolved because this is getting really confusing. It's a paradox after all. And I don't seem to be prepared to embrace the fact that standard mathematics are not enough to answer this.
The OP's sum is :
- divergent under standard tests.
- convergent using other methods.
How can a sum possibly be divergent and convergent ? There must be one of the methods used that fails. Unless I fail. But I'm going to give up because I feel I suck at this kind of mathematics. I'm not even able to understand this stuff even after reading twice the Wikipedia article
So the definition of the sum of a series as the limit of the partial sum is not actually correct? Or not always correct? I that sense, what we know about summation of series and all the tests we use are based on that definition. For the given series to converge, there must be another definition to use!