[seqfan] Re: help need with some Ramanujan identities

[Brad Klee]

In another thread we were talking about quartic plane curves, and I
mentioned an area
integral, which very slowly converges to Pi/sqrt(2). The situation here is
similar.
The signed g.f. of A099774 in powers x^(2*n),

G(x) = 1 - 2*x^2 + 2*x^4 - 2*x^6 + 3*x^8 +  . . .

diverges at x=1, while integral

F(x) = x - (2/3)*x^3 + (2/5)*x^5 - (2/7)*x^7 + (1/3)*x^9 + . . .

apparently converges to a finite value. The main difference seems to be
that G(x) is
not a differentiably finite function. However, there is at least one
well-known definition
for zeta(2) as a rational integral, i.e. a K-Z period (cf. [1] eq. 23 - 29
).

This question 770 is an interesting one, especially considering Part 2 [2],
on the
divergence of the signed g.f. of A000005 and its first two integrals,

G'(x) = 1 - 2*x + 2*x^2 - 3*x^3 + 2*x^4 - 4*x^5 . . .
F'(x) = x - x^2 + (2/3)*x^3 - (3/4)*x^4 + (2/5)*x^5 . . .
H'(x) = (1/2)*x^2 - (1/3)*x^3 + (1/6)*x^4/6 - (3/20)*x^5 . . .

Apparently F'(x) diverges to minus infinity while H'(x) converges to zero.
As the first
term of F'(x) is positive we find a root F'(x_0)=0 where
max(H'(x))=H'(x_0),

x_0 =   0.8361780481786675543697486454719680973158797361292534 . . .

The root x_0 is easy to approximate, even to hundreds of digits, and on the
first few
searches does not appear listed in OEIS.

Cheers,

Brad

[1] http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html
[2] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.9228 ( page
14 )
Berndt's article also references to the solutions to 770.


On Fri, Aug 31, 2018 at 3:26 AM Hugo Pfoertner <yae9911 at gmail.com> 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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