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Question Number 125857 by Dwaipayan Shikari last updated on 14/Dec/20

$$1+4{\left(\frac{1}{2}\right)}^7+7{\left(\frac{1.3}{2.4}\right)}^7+10{\left(\frac{1.3.5}{2.4.6}\right)}^7+...$$

Answered by Olaf last updated on 14/Dec/20

$$\mathrm{S}=1+\underset{n=1}{\sum ^{\mathrm{\infty }}}(1+3n){\left(\underset{k=1}{\prod ^n}\frac{(2k-1)}{\left(2k\right)}\right)}^7$$$$\mathrm{S}=1+\underset{n=1}{\sum ^{\mathrm{\infty }}}(1+3n){\left(\frac{1}{2^{n}n!}\underset{k=1}{\prod ^n}(2k-1)\right)}^7$$$$\mathrm{S}=1+\underset{n=1}{\sum ^{\mathrm{\infty }}}(1+3n){\left(\frac{1}{2^{n}n!}\underset{k=1}{\prod ^n}\frac{(2k-1)\left(2k\right)}{\left(2k\right)}\right)}^7$$$$\mathrm{S}=1+\underset{n=1}{\sum ^{\mathrm{\infty }}}(1+3n){\left(\frac{\left(2n\right)!}{2^{2n}n{!}^2}\right)}^7$$$$\mathrm{\Gamma }(n+\frac{1}{2})=\frac{\left(2n\right)!}{2^{2n}n!}\sqrt{\pi }$$$$\mathrm{S}=1+\frac{1}{{\pi }^{7/2}}\underset{n=1}{\sum ^{\mathrm{\infty }}}(1+3n){\left(\frac{\mathrm{\Gamma }(n+\frac{1}{2})}{\mathrm{\Gamma }(n+1)}\right)}^7$$$$...\mathrm{to}\mathrm{be}\mathrm{continued}...$$

Commented by Dwaipayan Shikari last updated on 14/Dec/20

$$Greatsir!$$

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