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Question Number 131795 by Dwaipayan Shikari last updated on 08/Feb/21

$$\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{5^3}+\frac{1}{{10}^3}+\frac{1}{{17}^3}+\frac{1}{{26}^3}+\frac{1}{{37}^3}+\frac{1}{{50}^3}+\frac{1}{{65}^3}+\frac{1}{{82}^3}+\frac{1}{{101}^3}+...$$

Answered by Olaf last updated on 08/Feb/21

$$\mathrm{Let}{u}_n=n^2-2n+2$$$$u_n=1,2,5,10,17,26,37,50,65...$$$$\mathrm{S}=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{{(n^2-2n+2)}^3}$$$$\mathrm{S}=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{{(n-e^{-i\frac{\pi }{4}})}^3{(n-e^{+i\frac{\pi }{4}})}^3}$$$$...$$

Commented by mindispower last updated on 08/Feb/21

$$=\sum _{n⩾1}\frac{1}{{({(n-1)}^2+1)}^3}$$$$=\sum _{n⩾0}\frac{1}{{(n^2+1)}^3}$$$$f\left(x\right)=\sum _{n⩾0}\frac{1}{n^2+x^2}$$$$cot\left(x\right)=\frac{1}{x}+2\mathrm{\Sigma }\frac{x}{x^2-n^2{\pi }^2}$$$$\Rightarrow \pi cot\left(\pi x\right)=\frac{1}{x}+2\mathrm{\Sigma }\frac{x}{x^2-n^2}$$$$x=it\Rightarrow \pi cot\left(i\pi t\right)=\frac{1}{it}+2\mathrm{\Sigma }\frac{it}{-t^2-n^2}$$$$i\pi th\left(\pi t\right)=\frac{1}{it}+2\sum _{n⩾1}\frac{it}{-t^2-n^2}$$$$\Rightarrow \sum _{n⩾1}\frac{1}{t^2+n^2}=\frac{\pi }{2t}th\left(\pi t\right)-\frac{1}{2t^2}$$$$\mathrm{\Sigma }\frac{1}{x+n^2}=\frac{\pi }{2\sqrt{x}}th\left(\pi \sqrt{x}\right)-\frac{1}{2x}$$$$tack3rdderivatandx=1$$$$$$

Commented by Dwaipayan Shikari last updated on 08/Feb/21

$$yessir!$$

Commented by mindispower last updated on 09/Feb/21

$$withepleasur$$

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