[seqfan] Re: help need with some Ramanujan identities

[israel at math.ubc.ca]

Modulo questions about rearrangement of non-absolutely convergent series, 
the result should indeed be (Pi/4)^2.

Sum_{k>=0} (-1)^k d(2k+1)/(2k+1) = Sum_{k>=0} Sum_{2i+1 | 2k+1} 
(-1)^k/(2k+1) (letting 2k+1=(2i+1)(2j+1): note that k == i+j mod 2)
  = Sum_{i>=0} Sum_{j>=0} (-1)^(i+j)/((2i+1)(2j+1))
  = (Sum_{i>=0} (-1)^i/(2i+1))^2 = (Pi/4)^2

Cheers,
Robert

On Aug 31 2018, Hugo Pfoertner wrote:

>Having seen Giovanni Resta's numerical results, summing 7.5*10^11 terms,
>one may really ask if https://oeis.org/A318590 = https://oeis.org/A222068
>and if therefore A318590 should be merged into A222068.
>
>On Wed, Aug 29, 2018 at 9:26 PM Hugo Pfoertner <yae9911 at gmail.com> wrote:
>
>> After summing 10^7 terms, I get 0.61685(016), which is a strange
>> coincidence with https://oeis.org/A222068 (Pi/4)^2=0.616850275...
>>
>> On Wed, Aug 29, 2018 at 8:21 PM Hugo Pfoertner <yae9911 at gmail.com> wrote:
>>
>>> To continue with Ramanujan's questions: Question 770 dealing with 
>>> alternating sums of d(n)=A000005 is said to have been solved in two 
>>> articles in the Journal of the Indian Mathematical Society. Ramanujan 
>>> had asked to show that the infinite sum d(1) - d(3)/3+ d(5)/5 - d(7)/7 
>>> + d(9)/9 - ... is a convergent series. Does somebody have access to the 
>>> articles or knows about a closed form solution? From a stupid summation 
>>> of 2*10^6 terms I get for the sum: 0.61684.., which is not in the OEIS, 
>>> so I created the draft https://oeis.org/draft/A318590
>>>
>>> Hugo Pfoertner
>>>
>>>
>
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