Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 117086 by mnjuly1970 last updated on 09/Oct/20

$$...nicemathematics...$$$$evaluate...$$$${lim}_{n\to \mathrm{\infty }}{\left(\underset{k=1}{\prod ^n}sin\left(\frac{k\pi }{4n}\right)\right)}^{\frac{1}{n}}=?$$$$$$$$...m.n.1970...$$$$$$

Answered by Ar Brandon last updated on 09/Oct/20

Commented by mnjuly1970 last updated on 09/Oct/20

$$thankyouverymuch..$$

Commented by Ar Brandon last updated on 09/Oct/20

You're welcome Sir ��

Answered by Dwaipayan Shikari last updated on 09/Oct/20

$$\underset{n\to \mathrm{\infty }}{\lim }{\left(\underset{k=1}{\prod ^n}sin\left(\frac{k\pi }{4n}\right)\right)}^{\frac{1}{n}}=y$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\underset{k=1}{\sum ^n}log\left(sin\left(\frac{k\pi }{4n}\right)\right)=logy$$$${\int }_0^{1}log\left(sin\left(\frac{\pi x}{4}\right)\right)=logy$$$$\frac{4}{\pi }{\int }_0^{\frac{\pi }{4}}log\left(sint\right)dt$$$$-1.27=logy\left(Wolfram\right)$$$$y=e^{-1.27}$$

Commented by Olaf last updated on 10/Oct/20

$${\int }_0^{\frac{\pi }{4}}\ln \left(\sin t\right)dt=-\frac{K}{2}-\frac{\pi }{4}\ln 2\mathrm{exactly}.$$$$(\mathrm{with}K=\beta \left(2\right):\mathrm{constant}\mathrm{of}\mathrm{Catalan})$$$$\mathrm{So}\mathrm{limit}\mathrm{is}\frac{1}{2}{e}^{-\frac{2\beta \left(2\right)}{\pi }}$$

Commented by mnjuly1970 last updated on 09/Oct/20

$$thankyousomuch.$$$$answer:=e^{(\frac{-2G}{\pi }-log\left(2\right))}✓$$$$recall:{\int }_0^{\frac{\pi }{4}}log\left(sin\left(x\right)\right)dx$$$$=\frac{-G}{2}-\frac{\pi }{4}log\left(2\right)$$$$m.n.1970$$$$sincerelyyours...$$$$$$

Commented by mnjuly1970 last updated on 10/Oct/20

$$youareright.$$$$yoursolutionis$$$$simplified...$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com