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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. – Max Muller May 5 '13 at 12:11
@MaxMuller Yes I did. The question was old and I have rephrased it. Why on MO? – Pedro Tamaroff May 5 '13 at 12:14
I suspect you have a higher chance to get this question answered if you post it on MO as well. – Max Muller May 5 '13 at 12:22
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? – Max Muller May 6 '13 at 13:13
Go ahead. ${}{}{}{}{}{}$ – Pedro Tamaroff May 6 '13 at 15:50
up vote 4 down vote accepted

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):

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. – Paramanand Singh Jul 3 '13 at 4:17

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