Clearly this diverges by nth term test. But what i'm not understanding is that the proofs provided on wikipedia make sense also! they even provide a proof to the following: !!! HOW IN THE WORLD CAN THIS BE EXPLAINED?

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- Apr 25th 2010, 10:34 AM #16

- Apr 25th 2010, 11:53 AM #17

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- Apr 25th 2010, 12:04 PM #18
What ? Now I must tell you that I am not convinced at all. Your expression is equivalent to :

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 ?

- Apr 25th 2010, 12:06 PM #19

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- Apr 25th 2010, 12:08 PM #20

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- Apr 25th 2010, 12:12 PM #21
Okay let's do this one quickly.

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.

- Apr 25th 2010, 12:17 PM #22

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We are not talking about the partial sums. See Divergent series - Wikipedia, the free encyclopedia

- Apr 25th 2010, 12:24 PM #23
Okay maddas, now I'm going to ask you something.

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

- Apr 25th 2010, 12:27 PM #24

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- Apr 25th 2010, 12:32 PM #25

- Apr 25th 2010, 12:41 PM #26

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- Apr 25th 2010, 01:05 PM #27
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!

- Apr 25th 2010, 01:11 PM #28

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- Apr 25th 2010, 02:05 PM #29

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- Apr 25th 2010, 02:06 PM #30

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