(1/3^3 )−(1/5^3 )+(1/(11^3 ))−(1/(13^3 ))+(1/(19^3 ))−(1/(21^3 ))+(1/(29^3 ))−(1/(31^3 ))+..


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Question Number 130748 by Dwaipayan Shikari last updated on 28/Jan/21

$\frac{1}{3^{3}} - \frac{1}{5^{3}} + \frac{1}{11^{3}} - \frac{1}{13^{3}} + \frac{1}{19^{3}} - \frac{1}{21^{3}} + \frac{1}{29^{3}} - \frac{1}{31^{3}} + . .$
	Answered by mindispower last updated on 28/Jan/21

	$= \sum_{k ⩾ 1} \frac{1}{{(8 k + 3)}^{3}} - \sum_{k ⩾ 1} \frac{1}{{(8 k - 3)}^{3}} + \frac{1}{3^{3}} . . . A$ $c o t (π x) = \frac{1}{x} + \sum_{n ⩾ 1} \frac{1}{(x - n)} + \frac{1}{(x + n)}$ $= \frac{1}{x} + \sum_{n ⩾ 1} (\frac{1}{x + n} + \frac{1}{x - n})$ $3 r d d e r i v a t e = \frac{2}{x^{3}} + Σ \frac{2}{{(x + n)}^{3}} - \frac{2}{{(n - x)}^{3}}$ $x = \frac{3}{8}, g i v e c l o s e f o r m e f o r A$
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