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It seems that under the light of Ramanujan Summation the following is plausible:

$$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$

Alas, I can't really find any concrete definition of Ramanujan Summation. Could someone provide a small explanation?

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    It seems you have modified your question ;) . At first it consisted of 4 sub-questions, right? By the way, perhaps it's a good idea to ask this question on mathoverflow.2013-05-05
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    @MaxMuller Yes I did. The question was old and I have rephrased it. Why on MO?2013-05-05
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    I suspect you have a higher chance to get this question answered if you post it on MO as well.2013-05-05
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    Are you going to post this question on MO as well? If not, may I post it there? If so, could you please let me know?2013-05-06
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    Go ahead. ${}{}{}{}{}{}$2013-05-06
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    Hmm perhaps it's wiser to ask a more focussed question on MO. The wikipedia article is quite elaborate. Could you please comment on which part(s) of the definition of the Ramanujan summation stated in that article elude you?2013-05-10
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    Were you right?2015-01-26

2 Answers 2

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Ramanujan's Theory of Summation is presented by Bruce C. Berndt in Ramanujan's Notebooks Vol 1, Chapter 6 titled "Ramanujan's Theory of Divergent Series".

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A Google search for "Ramanujan's Theory of Summation" gives this Wikipedia article (among others):

http://en.wikipedia.org/wiki/Ramanujan_summation

It states that "Ramanujan summation ..." takes "the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers...".

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    the poster has already provided this wikipedia link in the question.2013-07-03