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Question Number 75403 by TawaTawa last updated on 10/Dec/19

	Answered by mind is power last updated on 10/Dec/19
	$=(Σ_(m≥0) (1/((2m+1)^6 )))(Σ_(n=1) ^(+∞) (((−1)^(n+1) )/(4n^2 −1))) Σ_(n≥1) (((−1)^(n+1) )/((2n−1)(2n+1)))=Σ_(n≥1) (((−1)^(n+1) )/(4n^2 −1)) =(1/2){Σ_(n≥1) (((−1)^(n+1) )/((2n−1)))+Σ_(n≥1) (((−1)^(n+2) )/(2n+1))} =(1/2)+{Σ_(n≥1) (((−1)^(n+2) )/(2n+1))} =(1/2)+Σ_(n≥1) (((−1)^n )/(2n+1)) (1/(1+x^2 ))=Σ_(k≥0) (−x^2 )^k ⇒∫(1/(1+x^2 ))=Σ_(k≥0) (((−1)^k )/(2k+1))x^(2k+1) ⇒Σ_(k≥0) (((−1)^k )/(2k+1))=arctan(1)=(π/4) ⇒Σ_(k≥1) (((−1)^k )/(2k+1))=(π/4)−1 ⇒Σ_(n≥1) (1/(4n^2 −1))=(π/4)−(1/2)=((π−2)/4) ζ(6)=Σ_(k≥1) (1/k^6 )=(π^6 /(945)) ⇒Σ_(k≥1) (1/((2k+1)^6 ))=ζ(6)−(1/2^6 )ζ(6)=(((2^6 −1)/2^6 )).(π^6 /(945))=((63π^6 )/(64.945))=(π^6 /(960)) ⇒Σ_(m≥0) Σ_(n≥1) (1/((2m+1)^6 )).(((−1)^(n+1) )/(4n^2 −1))=(π^6 /(960)).((π−2)/4)=((π^6 (π−2))/(3840))$
	$= (\sum_{m ⩾ 0} \frac{1}{{(2 m + 1)}^{6}}) (\underset{n = 1}{\sum^{+ \infty}} \frac{{(- 1)}^{n + 1}}{{4 n}^{2} - 1})$ $\sum_{n ⩾ 1} \frac{{(- 1)}^{n + 1}}{(2 n - 1) (2 n + 1)} = \sum_{n ⩾ 1} \frac{{(- 1)}^{n + 1}}{{4 n}^{2} - 1}$ $= \frac{1}{2} {\sum_{n ⩾ 1} \frac{{(- 1)}^{n + 1}}{(2 n - 1)} + \sum_{n ⩾ 1} \frac{{(- 1)}^{n + 2}}{2 n + 1}}$ $= \frac{1}{2} + {\sum_{n ⩾ 1} \frac{{(- 1)}^{n + 2}}{2 n + 1}}$ $= \frac{1}{2} + \sum_{n ⩾ 1} \frac{{(- 1)}^{n}}{2 n + 1}$ $\frac{1}{1 + x^{2}} = \sum_{k ⩾ 0} {(- x^{2})}^{k} \Rightarrow \int \frac{1}{1 + x^{2}} = \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{2 k + 1} x^{2 k + 1}$ $\Rightarrow \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{2 k + 1} = \arctan (1) = \frac{π}{4}$ $\Rightarrow \sum_{k ⩾ 1} \frac{{(- 1)}^{k}}{2 k + 1} = \frac{π}{4} - 1$ $\Rightarrow \sum_{n ⩾ 1} \frac{1}{{4 n}^{2} - 1} = \frac{π}{4} - \frac{1}{2} = \frac{π - 2}{4}$ $ζ (6) = \sum_{k ⩾ 1} \frac{1}{k^{6}} = \frac{π^{6}}{945}$ $\Rightarrow \sum_{k ⩾ 1} \frac{1}{{(2 k + 1)}^{6}} = ζ (6) - \frac{1}{2^{6}} ζ (6) = (\frac{2^{6} - 1}{2^{6}}) . \frac{π^{6}}{945} = \frac{63 π^{6}}{64.945} = \frac{π^{6}}{960}$ $\Rightarrow \sum_{m ⩾ 0} \sum_{n ⩾ 1} \frac{1}{{(2 m + 1)}^{6}} . \frac{{(- 1)}^{n + 1}}{{4 n}^{2} - 1} = \frac{π^{6}}{960} . \frac{π - 2}{4} = \frac{π^{6} (π - 2)}{3840}$
	Commented by TawaTawa last updated on 10/Dec/19

	$God bless you sir, i appreciate your time$
	Commented by mind is power last updated on 10/Dec/19

	$y^{'} re Welcom$
	Commented by feli last updated on 11/Dec/19

	$p l s h o w c a n i c o n t a c t y o u o n w h a t s a p p s i r m i n d i s p o w e r$
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