........advanced... ... ...math...... prove that _∗^∗ :::: 𝛀=Σ_(k=0) ^∞ {(1/(16^k ))((4/(8k+1))−(2/(8k+4))−(1/(8k+5))−(1/(8k+6)))}=π ....Bailey−Borwein formula....


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Question Number 138723 by mnjuly1970 last updated on 17/Apr/21
$........advanced... ... ...math...... prove that _∗^∗ :::: 𝛀=Σ_(k=0) ^∞ {(1/(16^k ))((4/(8k+1))−(2/(8k+4))−(1/(8k+5))−(1/(8k+6)))}=π ....Bailey−Borwein formula....$
$. . . . . . . . a d v a n c e d . . . . . . . . . m a t h . . . . . .$ $p r o v e t h a t_{}^{} ::::$ $Ω = \underset{k = 0}{\sum^{\infty}} {\frac{1}{16^{k}} (\frac{4}{8 k + 1} - \frac{2}{8 k + 4} - \frac{1}{8 k + 5} - \frac{1}{8 k + 6})} = π$ $. . . . B a i l e y - B o r w e i n f o r m u l a . . . .$
Commented by Dwaipayan Shikari last updated on 17/Apr/21

$I t i s a s h a r d a s t o p r o v e \frac{1}{π} = \frac{2 \sqrt{2}}{99^{2}} \underset{n = 0}{\sum^{\infty}} \frac{(1103 + 26390 n) (4 n)!}{{(n!)}^{4} 396^{n}}; ($
Commented by mnjuly1970 last updated on 17/Apr/21

$y e s y o u a r e r i g h t . . .$
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