Ramanujan Summation

Question

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?

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 Lonysa Muller, May 05 '13 at 12:11
@MaxMuller Yes I did. The question was old and I have rephrased it. Why on MO? — Pedro, May 05 '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 Lonysa Muller, May 05 '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 Lonysa Muller, May 06 '13 at 13:13
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? — Max Lonysa Muller, May 10 '13 at 14:30

score 9 · Accepted Answer · answered Jul 02 '13 at 15:29

9

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".

answered Jul 02 '13 at 15:29

Paramanand Singh

93,051

score 3 · Answer 2 · answered Jul 02 '13 at 16:22

3

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...".

answered Jul 02 '13 at 16:22

marty cohen

110,916

4

the poster has already provided this wikipedia link in the question. – Paramanand Singh Jul 03 '13 at 04:17
1

The Wiki article is of very poor quality. Currently, the first sentence of the first section reads, "Since there are no properties of an entire sum, the Ramanujan summation functions as a property of partial sums." This is incomprehensible to me. – WillG Jul 29 '23 at 16:07

score 0 · Answer 3 · answered Jun 13 '19 at 13:32

0

The Ramanujan summation does a partial sum on a normally divergent series. It simply assigns a "meaningful value" to an otherwise divergent series. It should not normally be used on a convergent series.

answered Jun 13 '19 at 13:32

Highvoltagemath

529

Ramanujan Summation

3 Answers3