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Before we indulge in the facts, let's munch on some pie. (pun intended)
Take The Nilakantha Series and The Gregory-Leibniz Series.
It can be transformed into
where is the Riemann Zeta function and so that the error after terms is
There is also Machin's Formula:
Abraham Sharp gave the infinite sum series,
Simple series' of infinite sums:
There is also
In 1666 (Newton's miracle year) Newton used a geometric construction to derive the formula
The coefficients can be found in
by taking the series expansion of about 0 obtaining
Euler's convergence improvement transformation gives
This corresponds to plugging in into the power series for the hypergeometric function
Despite the convergence improvement, series converges at only one bit/term. At the cost of a square root, Gosper has noted that gives 2 bits/term,
and gives almost 3.39 bits/term
where is the golden ratio (not Euler's Totient function). Gosper also obtained
A spigot algorithm for is given by Rabinowitz and Wagon.
More amazingly still, a closed-form expression giving a digit-extraction algorithm which produces digits of (or in base-16 was discovered by Bailey.
This formula, known as the BBP formula, was discovered using the PSLQ algorithm and is equivalent to
There is a series of BBP-type formulas for in powers of , the first few independent formulas of which are
Similarly, there are a series of BBP-type formulas for in powers of , the first few independent formulas of which are
F. Bellard found the rapidly converging BBP-type formula
A related integral is
Boros and Moll state that it is not clear if there exists a natural choice of a rational polynomial whose integral between 0 and 1 produces , where is the next convergent. However, an integral exists for the fourth convergent, namely
Backhouse used the identity
for positive integer and and where and are rational constant to generate a number of formulas for . In particular, if then
A similar formula was subsequently discovered by Ferguson, leading to a two-dimensional lattice of such formulas which can be generated by these two formulas given by
The Wallis Product is another magnificent way of expressing :
1)Euler's infinite product for the sine function
2)Proof using Integration
Integrate by parts:
There is also a continued fraction form :
There is also a curious identity where involving the first digits of
I am going to update this in the future and in the meantime, we can comment about the magical number,
Watch Out! There is also Golden ratio in another note!
Sources: Mathworld Wolfram and Wikipedia
Help from Members: Agnishom Chattopadhyay, X X, Andrei Li
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There is also an amazing song out there!