Question Number 143047 by 0731619 last updated on 09/Jun/21 Answered by JDamian last updated on 09/Jun/21 $$\Sigma\frac{\mathrm{1}}{{k}^{\mathrm{3}} }\:\:\mathrm{is}\:\mathrm{unknown} \\ $$ Commented by ArielVyny last updated…
Question Number 143045 by 0731619 last updated on 09/Jun/21 Answered by Dwaipayan Shikari last updated on 09/Jun/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}^{{n}} }{\mathrm{1}−{x}}{dx}=−\gamma+\psi\left({n}+\mathrm{1}\right) \\ $$$$\underset{{n}\rightarrow\mathrm{3}} {\mathrm{lim}}\frac{−\gamma−\mathrm{1}.\mathrm{833}+\psi\left({n}+\mathrm{1}\right)}{\Gamma\left({n}+\mathrm{1}\right)−\mathrm{6}}\overset{{Lhopital}} {=}\frac{\psi'\left({n}+\mathrm{1}\right)}{\Gamma'\left({n}+\mathrm{1}\right)}=\frac{\psi'\left(\mathrm{4}\right)}{\psi\left(\mathrm{4}\right)\Gamma\left(\mathrm{4}\right)}…
Question Number 143046 by 0731619 last updated on 09/Jun/21 Answered by mindispower last updated on 09/Jun/21 $${x}^{{x}^{{x}} } ={f}\left({x}\right)\Rightarrow{ln}\left({fx}\right)={f}\left({x}\right){ln}\left({x}\right) \\ $$$${ln}\left({f}\left({x}\right)\right){e}^{−{ln}\left({f}\left({x}\right)\right)} ={ln}\left({x}\right) \\ $$$$\Rightarrow−{ln}\left({f}\left({x}\right)\right)={W}\left(−{ln}\left({x}\right)\right) \\…
Question Number 143043 by 0731619 last updated on 09/Jun/21 Answered by Dwaipayan Shikari last updated on 09/Jun/21 $${y}={x}!^{{x}!} \\ $$$${log}\left({y}\right)={x}!{log}\left({x}!\right) \\ $$$${log}\left({y}\right)=\Gamma\left({x}+\mathrm{1}\right){log}\left(\Gamma\left({x}+\mathrm{1}\right)\right) \\ $$$$\frac{{y}'}{{y}}=\Gamma'\left({x}+\mathrm{1}\right){log}\left(\Gamma\left({x}+\mathrm{1}\right)\right)+\Gamma'\left({x}+\mathrm{1}\right) \\…
Question Number 77504 by Maclaurin Stickker last updated on 07/Jan/20 Answered by mr W last updated on 07/Jan/20 $${say}\:{the}\:{radius}\:{of}\:{large}\:{outer}\:{circle}={R} \\ $$ Commented by mr W…
Question Number 143042 by 0731619 last updated on 09/Jun/21 Answered by Dwaipayan Shikari last updated on 09/Jun/21 $$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{log}\left({x}\right)={x}−\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{log}\left({x}\right)\frac{{x}^{{x}^{{x}^{?…} } } }{{x}^{\mathrm{2}}…
Question Number 143032 by ZiYangLee last updated on 09/Jun/21 $$\:\:\:\:\:\int\:\mathrm{sin}^{−\mathrm{5}} {x}\:{dx}\:=? \\ $$ Answered by Olaf_Thorendsen last updated on 09/Jun/21 $$\mathrm{F}\left({x}\right)\:=\:\int\frac{{dx}}{\mathrm{sin}^{\mathrm{5}} {x}} \\ $$$$\mathrm{Let}\:{t}\:=\:\mathrm{tan}\frac{{x}}{\mathrm{2}} \\…
Question Number 11956 by Mahmoud A.R last updated on 07/Apr/17 $${A}\:,\:\mathrm{2}{B}\:,\mathrm{3}{C}\:,\mathrm{4}{D}\: \\ $$$${are}\:{positive}\:{numbers}\:{forming}\:{a}\: \\ $$$${geometric}\:{series}\: \\ $$$${prov}\:{that}\:: \\ $$$$\left({A}\:+\:\mathrm{3}{C}\right)\:\left({B}\:+\:\mathrm{2}{D}\right)\:>\:\mathrm{2} \\ $$ Commented by ajfour last…
Question Number 143031 by rs4089 last updated on 09/Jun/21 Answered by mnjuly1970 last updated on 09/Jun/21 $$\frac{\pi^{\mathrm{2}} }{\mathrm{6}}+{ln}^{\mathrm{2}} \left(\mathrm{2}\right)…. \\ $$ Answered by Dwaipayan Shikari…
Question Number 143027 by mathlove last updated on 09/Jun/21 Commented by Rasheed.Sindhi last updated on 09/Jun/21 $${x}+\frac{\mathrm{1}}{{x}}=−\mathrm{1}\Rightarrow{x}=\omega,\omega^{\mathrm{2}} \\ $$$${When}\:{x}=\omega \\ $$$${x}^{\mathrm{123456789}} +\frac{\mathrm{1}}{{x}^{\mathrm{123456789}} } \\ $$$$\:\:\:\:\:=\omega^{\mathrm{123456789}}…