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Author: Tinku Tara

n-1-1-n-

Question Number 219098 by zetamaths last updated on 20/Apr/25 $$\zeta\left(\alpha\right)=\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\alpha} }\:\: \\ $$ Answered by Frix last updated on 20/Apr/25 $$\mathrm{This}\:\mathrm{is}\:\mathrm{just}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:\mathrm{the}\:\zeta−\mathrm{Function}, \\ $$$$\mathrm{there}'\mathrm{s}\:\mathrm{nothing}\:\mathrm{to}\:\mathrm{solve}.…

Question-219093

Question Number 219093 by hardmath last updated on 19/Apr/25 Commented by hardmath last updated on 19/Apr/25 $$\mathrm{ABC}\:=\:\bigtriangleup \\ $$$$\mathrm{B}\:-\:\mathrm{acute}\:\mathrm{angle} \\ $$$$\angle\mathrm{B}\:=\:\mathrm{2}\:\centerdot\:\angle\mathrm{C} \\ $$$$\mathrm{AB}\:=\:\mathrm{10} \\ $$$$\mathrm{BC}\:=\:\mathrm{22}…

Prove-that-the-sequence-a-n-1-n-1-n-is-decreasing-

Question Number 219090 by depressiveshrek last updated on 19/Apr/25 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:{a}_{{n}} =\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{n}!}}\:\mathrm{is}\:\mathrm{decreasing}. \\ $$ Answered by Frix last updated on 19/Apr/25 $${n}\rightarrow\infty\:\Rightarrow\:{n}!\approx\left(\frac{{n}}{\mathrm{e}}\right)^{{n}} \sqrt{\mathrm{2}\pi{n}}\:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{n}!}}\approx\frac{\mathrm{e}}{\left(\mathrm{2}\pi{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}{x}}} {x}}…

Question-219087

Question Number 219087 by fantastic last updated on 19/Apr/25 Answered by mahdipoor last updated on 19/Apr/25 $${DE}=\mathrm{2}{r}−{AD}−{EB}=\mathrm{2}{r}−\left({d}\right)−\left({r}−{d}\right)={r} \\ $$$$\frac{\pi}{\mathrm{2}}\left(\left(\mathrm{2}{r}\right)^{\mathrm{2}} −\left({r}\right)^{\mathrm{2}} \right)=\mathrm{1}.\mathrm{5}\pi{r}^{\mathrm{2}} \\ $$ Commented by…