# Re: Stepen Lucas's "Integral approximations to Pi with nonnegative integrands"

*From*: "Alexander R.Povolotsky" <apovolot@xxxxxxxxx>*Date*: Sat, 21 Feb 2009 15:08:50 -0800 (PST)

On Feb 19, 7:49 pm, "Alexander R.Povolotsky" <apovo...@xxxxxxxxx>

wrote:

Hi,

S. K. Lukas in

http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf>

derived several identities, which relate Pi (via definite integrals)

with

the several few Pi fractional convergents, which denominators and

numerators

are described in

http://www.research.att.com/~njas/sequences/A002486<http://www.research.att.com/%7Enjas/sequences/A0024865>http://www.research.att.com/~njas/sequences/A002485<http://www.research.att.com/%7Enjas/sequences/A0024865>

I raised the issue re the possibility of deriving generalized ("n"

parameter

based ) definite integral identity relating Pi with ALL (each at its

own

value of n) Pi fractional convergents (referenced in above sequences)

- see my exchange with S. K. Lukas below.

Unfortunately I do not have sufficient computational resources (I do

not have access to Maple or Mathematica, instead I have Pari/GP

installed

on my very old home computer) to take advantage of Stephen's generous

offer to play with his Maple program, which he wrote and which is

listed in

http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf>

The program is on page 9 of the linked to pdf and

that it is literally only 18 lines of code.

May one of you could help me running

Stephen's program towards experimental attempt of deriving desired

generalized

identity?

Thanks,

Best Regards,

Alexander R. Povolotsky

May be the search should be based around finding

i(n), k(n), l(n) and m(n) as some integer functions of n, where i(n)

appears to represent the various powers of 2 ) in the following (using

Maple's notations) guessed (by me ) identity

Pi = A002485(n)/A002486(n) - 1/(i*l)*Int(x^m*(1-x)^m*(k+(k+l)*x^2)/

(1+x^2),x = 0 .. 1)

Does above guesstimate help in writing a program ?

Of course my above identity guess may be not accurate and it may be

that in the end the actual experimental finding might reveal something

else (or nothing at all ;-) )

Cheers,

Alex (Alexander R. Povolotsky )

.

**References**:**Stepen Lucas's "Integral approximations to Pi with nonnegative integrands"***From:*Alexander R.Povolotsky

- Prev by Date:
**Symbolic matrix operations in mathematica** - Next by Date:
**Re: simply simplify this, a sin-ful expression** - Previous by thread:
**Stepen Lucas's "Integral approximations to Pi with nonnegative integrands"** - Next by thread:
***** cheap Discount Tiffany & co online store mytiffanycity.com/** - Index(es):