Pi Approximation Day
07-25-2019, 08:14 PM (This post was last modified: 07-25-2019 08:35 PM by Gerson W. Barbosa.)
Post: #22
 Gerson W. Barbosa Senior Member Posts: 1,452 Joined: Dec 2013
RE: Pi Approximation Day
(07-24-2019 10:30 AM)BartDB Wrote:
(07-23-2019 06:18 PM)Gerson W. Barbosa Wrote:  That is,

π = 22/7 - ∫(0,1,X^4*(1-X)^4/(1+X^2),X)

Similar equalities can be automatically obtained on the HP-50g with help of a small User-RPL program. The Egyptian Fractions part --

{ }
WHILE SWAP DUP -5. ALOG SQ >
REPEAT DUP INV CEIL DUP UNROT INV - UNROT +
END 6. ALOG SQ * SWAP

-- is borrowed code from forumer 3298 here.

For example,

22 ENTER 7

\<< / \->NUM DUP IP R\->I DUP UNROT - { }
WHILE SWAP DUP -5. ALOG SQ >
REPEAT DUP INV CEIL DUP UNROT INV - UNROT +
END 6. ALOG SQ * SWAP NIP DUP SIZE NOT NOT
{ 1 - X SWAP ^ 0 + \GSLIST + } { DROP } IFTE
4 X 2 ^ 1 + / - COLLECT
\>>

EVAL

--> '(X^8+X^6+3*X^2-1)/(X^2+1)'

Indeed,

'∫(0,1,(X^8+X^6+3*X^2-1)/(X^2+1),X)'

EVAL DISTRIB

--> '-π+22/7'

That is,

π = 22/7 - ∫(0,1,(X^8+X^6+3*X^2-1)/(X^2+1),X)

Notice this is a different integrand polynomial. The original one is more elaborate so that the difference area is continuous, not distributed between both sides of the x-axis.

Likewise,

π = 377/120 - ∫(0,1,(X^61+X^59+X^9+X^7+3*X^2-1)/(X^2+1),X)

and

π = 3 + ∫(0,1,(-3*X^2+1)/(X^2+1),X)
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 Messages In This Thread Pi Approximation Day - Gerson W. Barbosa - 07-22-2019, 03:41 AM RE: Pi Approximation Day - ggauny@live.fr - 07-22-2019, 10:01 AM RE: Pi Approximation Day - burkhard - 07-22-2019, 01:02 PM RE: Pi Approximation Day - rprosperi - 07-22-2019, 01:05 PM RE: Pi Approximation Day - Gerson W. Barbosa - 07-22-2019, 03:14 PM RE: Pi Approximation Day - Gerson W. Barbosa - 07-22-2019, 05:19 PM RE: Pi Approximation Day - Gerson W. Barbosa - 08-14-2019, 09:14 PM RE: Pi Approximation Day - Albert Chan - 08-14-2019, 10:46 PM RE: Pi Approximation Day - Gerson W. Barbosa - 08-14-2019, 11:09 PM RE: Pi Approximation Day - Albert Chan - 08-15-2019, 03:38 AM RE: Pi Approximation Day - Claudio L. - 07-22-2019, 07:48 PM RE: Pi Approximation Day - Dave Shaffer - 07-23-2019, 05:07 PM RE: Pi Approximation Day - ijabbott - 07-23-2019, 05:18 PM RE: Pi Approximation Day - Gerson W. Barbosa - 07-24-2019, 12:35 AM RE: Pi Approximation Day - jebem - 07-24-2019, 06:03 AM RE: Pi Approximation Day - bshoring - 07-22-2019, 09:40 PM RE: Pi Approximation Day - Valentin Albillo - 07-22-2019, 10:49 PM RE: Pi Approximation Day - Gerson W. Barbosa - 07-23-2019, 03:15 PM RE: Pi Approximation Day - BartDB - 07-23-2019, 05:31 PM RE: Pi Approximation Day - Gerson W. Barbosa - 07-23-2019, 06:18 PM RE: Pi Approximation Day - BartDB - 07-24-2019, 10:30 AM RE: Pi Approximation Day - Gerson W. Barbosa - 07-24-2019, 12:37 PM RE: Pi Approximation Day - Gerson W. Barbosa - 07-25-2019 08:14 PM RE: Pi Approximation Day - ijabbott - 07-23-2019, 05:09 PM RE: Pi Approximation Day - Erwin - 07-23-2019, 08:26 PM RE: Pi Approximation Day - Albert Chan - 07-25-2019, 01:26 PM RE: Pi Approximation Day - Bill Duncan - 07-26-2019, 11:02 PM RE: Pi Approximation Day - Gerson W. Barbosa - 08-20-2019, 04:10 PM RE: Pi Approximation Day - Leviset - 08-14-2019, 09:36 PM

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