Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 144607 by aliibrahim1 last updated on 26/Jun/21

Answered by mathmax by abdo last updated on 27/Jun/21

$${\mathrm{A}}_{\mathrm{n}}=\prod _{\mathrm{k}=2}^{\mathrm{n}}\mathrm{e}{(1-\frac{1}{{\mathrm{k}}^2})}^{{\mathrm{k}}^2}\Rightarrow \prod _{\mathrm{n}=2}^{\mathrm{\infty }}(....)={\lim }_{\mathrm{n}⌣\mathrm{\infty }}{\mathrm{A}}_{\mathrm{n}}$$$${\mathrm{A}}_{\mathrm{n}}={\mathrm{e}}^{\mathrm{n}-1}\prod _{\mathrm{k}=2}^{\mathrm{n}}{(1-\frac{1}{{\mathrm{k}}^2})}^{{\mathrm{k}}^2}\Rightarrow \log \left({\mathrm{A}}_{\mathrm{n}}\right)=\mathrm{n}-1+\sum _{\mathrm{k}=2}^{\mathrm{n}}{\mathrm{k}}^2\log (1-\frac{1}{{\mathrm{k}}^2})$$$${\log }^{{}^{\prime }}(1-\mathrm{x})=-\frac{1}{1-\mathrm{x}}=-(1+\mathrm{x}+{\mathrm{x}}^2+\mathrm{o}\left({\mathrm{x}}^3\right))\Rightarrow$$$$\log (1-\mathrm{x})\sim -\mathrm{x}-\frac{{\mathrm{x}}^2}{2}\Rightarrow \log (1-\frac{1}{{\mathrm{k}}^2})\sim -\frac{1}{{\mathrm{k}}^2}-\frac{1}{{2\mathrm{k}}^4}\Rightarrow$$$${\mathrm{k}}^2\log (1-\frac{1}{{\mathrm{k}}^2})\sim -1-\frac{1}{{2\mathrm{k}}^2}\Rightarrow \sum _{\mathrm{k}=2}^{\mathrm{n}}{\mathrm{k}}^2\log (...)\sim -(\mathrm{n}-1)$$$$-\frac{1}{2}\sum _{\mathrm{k}=2}^{\mathrm{n}}\frac{1}{{\mathrm{k}}^2}\Rightarrow \log \left({\mathrm{A}}_{\mathrm{n}}\right)\sim -\frac{1}{2}\sum _{\mathrm{k}=2}^{\mathrm{n}}\frac{1}{{\mathrm{k}}^2}$$$$=-\frac{1}{2}({\xi }_{\mathrm{n}}\left(2\right)-1)=\frac{1}{2}-\frac{{\xi }_{\mathrm{n}}\left(2\right)}{2}\Rightarrow$$$${\lim }_{\mathrm{n}\to +\mathrm{\infty }}\log \left({\mathrm{A}}_{\mathrm{n}}\right)=\frac{1}{2}-\frac{{\pi }^2}{12}\Rightarrow {\lim }_{\mathrm{n}\to +\mathrm{\infty }}{\mathrm{A}}_{\mathrm{n}}={\mathrm{e}}^{\frac{1}{2}-\frac{{\pi }^2}{12}}$$$$$$

Commented by aliibrahim1 last updated on 27/Jun/21

$$thxsir$$

Commented by mathmax by abdo last updated on 28/Jun/21

$$\mathrm{you}\mathrm{are}\mathrm{welcome}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com