...nice calculus... prove that:: (2/3)((1/4)−(1/(20))+(1/(56))−(1/(120))+(1/(220))−...) =^(???) π −3


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Question Number 129084 by mnjuly1970 last updated on 12/Jan/21

$. . . n i c e c a l c u l u s . . .$ $p r o v e t h a t ::$ $\frac{2}{3} (\frac{1}{4} - \frac{1}{20} + \frac{1}{56} - \frac{1}{120} + \frac{1}{220} - . . .)$ $\overset{? ? ?}{=} π - 3$
	Answered by Dwaipayan Shikari last updated on 12/Jan/21

	$\frac{1}{6} (1 - \frac{1}{5} + \frac{1}{14} - \frac{1}{30} + \frac{1}{55} - . .)$ $= \frac{1}{6} (1 - \frac{1}{1^{2} + 2^{2}} + \frac{1}{1^{2} + 2^{2} + 3^{2}} - \frac{1}{1^{2} + 2^{2} + 3^{2} + 4^{2}} - . . .)$ $= \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{n (n + 1) (2 n + 1)} = π - 3$
	Commented by mnjuly1970 last updated on 12/Jan/21

	$e x c e l l e n t m r p a y a n .$ $t h a n k s a l o t . . .$
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