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Yet another π formula
01-09-2021, 09:22 PM (This post was last modified: 11-06-2021 04:21 PM by Albert Chan.)
Post: #14
RE: Yet another π formula
Hi, Gerson W. Barbosa

I don't know how to quantify "optimal" correction. May be you want to proof this instead ?

\(\pi = \displaystyle\left( 2 + 4\sum_{k=1}^{n}\frac{(-1)^{k-1}}{4k^{2}-1} \right )
+ \lim_{k\rightarrow \infty }\left(\frac{(-1)^{n}} {b\;-\;\frac{3}{\large b+8\;-\;
\frac{60}{\frac{\ddots }{b+4k(k-1)\;-\;\frac{k^2 (4k^2-1)}{ b+4k(k+1) }}}}}\right)
,\quad b = 2n(n+1) + {3\over2}\)


This is not a proof, but I am convinced that above is true. Nice work Smile

First, try n=0, i.e. summation term = 0.
Relied only on limit, convergence rate is bad, but it is indeed approach pi.

>>> from gmpy2 import *
>>> get_context().precision = 1000 # ~ 300 dec. digits
>>> pi = const_pi()
>>> for i in range(7): print i, format(pi - test_pi(n = 0, cf = 10**i), 'g')
...
0  0.297148
1  0.0675375
2  0.00772836
3  0.000784124
4  7.85271e-05
5  7.85385e-06
6  7.85397e-07

Now, try combination of summation and limit.

>>> for i in range(7): print i, format(pi - test_pi(n = 9, cf = 10**i), 'g')
...
0  -5.30332e-21
1  -1.07719e-25
2  -1.47115e-39
3  -8.06555e-58
4  -9.64229e-77
5  -9.81688e-96
6  -9.83452e-115

>>> for i in range(7): print i, format(pi - test_pi(n = 10, cf = 10**i), 'g')
...
0  4.18442e-23
1  5.76716e-28
2  6.21078e-43
3  4.82312e-63
4  5.98619e-84
5  6.11764e-105
6  6.13095e-126

Note: test_pi() from previous post, modified to allow extra cf terms, beyond n.
Note: unlike summation term, CF correction convergence is one-sided.
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Messages In This Thread
Yet another π formula - Gerson W. Barbosa - 01-04-2021, 08:41 PM
RE: Yet another π formula - Albert Chan - 01-05-2021, 10:50 PM
RE: Yet another π formula - Albert Chan - 01-06-2021, 01:32 AM
RE: Yet another π formula - Albert Chan - 01-07-2021, 09:56 PM
RE: Yet another π formula - toml_12953 - 01-06-2021, 02:10 AM
RE: Yet another π formula - ttw - 01-06-2021, 03:44 AM
RE: Yet another π formula - Albert Chan - 01-09-2021 09:22 PM
RE: Yet another π formula - Albert Chan - 11-06-2021, 06:28 PM



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