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Expansiones de PI
(Mathemática | Cabos sueltos | Constantes | PI)

Descubridor: Arquímedes (287-212 BC) averiguó que 3 10/71 < PI < 3 1/7

PI = 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...

La fórmula de Vieta

2/PI = sqrt2/2 * sqrt( 2 + sqrt2 )/2 * sqrt(2 + ( sqrt( 2 + sqrt2) ) )/2 * ...c

La fómula de Leibnitz

PI/4 = 1/1 - 1/3 + 1/5 - 1/7 + ...

El producto de Wallis

PI/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...

2/PI = (1 - 1/2^2)(1 - 1/4^2)(1 - 1/6^2)...

La fórmula de Lord Brouncker


4/PI = 1 +        1

           ----------------

           2 +     3^2

               ------------

               2 +   5^2

                  ---------

                  2 + 7^2 ...

(PI^2)/8 = 1/1^2 + 1/3^2 + 1/5^2 + ...

(PI^2)/24 = 1/2^2 + 1/4^2 + 1/6^2 + ...

La fórmula de Euler

(PI^2)/6 = SUM (n = 1..inf) 1/n^2 = 1/1^2 + 1/2^2 + 1/3^2 + ...

(o mas generalmente...)

SUM (n = 1..inf) 1/n^(2k) = (-1)^(k-1) PI^(2k) 2^(2k) B(2k) / ( 2(2k)!)

B(k) = el k th Número Bernoulli. ej. B0=1 B1=-1/2 B2=1/6 B4=-1/30 B6=1/42 B8=-1/30 B10=5/66. Números Bernoulli adicionales se definen como (n 0)B0 + (n 1)B1 + (n 2)B2 + ... + (n (n-1))B(N-1) = 0 presumiendo que todos los Números impares de Bernoulli >1 son = 0. (n k) = coeficiente de binómico = n!/(k!(n-k)!)


  
 
  

 
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