Question Number 139479 by mnjuly1970 last updated on 27/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{advanced}\:….\bigstar\bigstar\bigstar…..{calculus}….. \\ $$$$\:\:\:\:\:\:\:\:\:\:\: :=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{{sin}\left({n}\right)}{{n}}\right)^{\mathrm{3}} =?\: \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$ Answered by Dwaipayan Shikari last…
Question Number 139457 by mnjuly1970 last updated on 27/Apr/21 $$\:_{} \:\:\: \\ $$$$\:\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}{n}\right)!}\:=\frac{{e}}{\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{3}\sqrt{{e}}}\:{cos}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right) \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…
Question Number 139404 by mnjuly1970 last updated on 26/Apr/21 $$\:\:\:\:\: \\ $$$$\:\:\:#\:{Question}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{proof}\:::\:\underset{{n}=−\infty} {\overset{\:\infty} {\sum}}\left(\:\frac{{sin}\left({n}\right)}{{n}}\:\right)^{\mathrm{2}} =\pi \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:…….\:{nice}\:.{calculus}\:……\:\:\:\:\: \\ $$ Answered by Dwaipayan Shikari…
Question Number 139323 by snipers237 last updated on 25/Apr/21 $$\:{Let}\:{f}\:{define}\:{such}\:{as}\:\:{f}\left(\mathrm{1}\right)=\mathrm{1},{f}\left(\mathrm{3}\right)=\mathrm{3} \\ $$$$\forall\:{n}\geqslant\mathrm{2}\:\:,\:{f}\left(\mathrm{2}{n}\right)={f}\left({n}\right)\: \\ $$$${f}\left(\mathrm{4}{n}+\mathrm{1}\right)=\mathrm{2}{f}\left(\mathrm{2}{n}+\mathrm{1}\right)−{f}\left({n}\right) \\ $$$${f}\left(\mathrm{4}{n}+\mathrm{3}\right)=\mathrm{3}{f}\left(\mathrm{2}{n}+\mathrm{1}\right)−\mathrm{2}{f}\left({n}\right) \\ $$$$ \\ $$$$\left.\mathrm{1}\right){Prove}\:{that}\:\forall\:{n}\:,\:{f}\left({n}\right)\:{is}\:{odd} \\ $$$$\left.\mathrm{2}\right){Prove}\:{that}\:{if}\:\:\:{f}\left({a}_{{n}} \right)={a}_{{n}} \:, \\…
Question Number 8244 by lepan last updated on 04/Oct/16 $${Show}\:{that}\:{the}\:{curve}\:{y}={ln}\left(\frac{\mathrm{5}−\mathrm{7}{x}}{\mathrm{8}+{x}}\right)\:{has} \\ $$$${no}\:{stationary}\:{point}\:{for}\:{all}\:{real}\:{values} \\ $$$${of}\:{x}. \\ $$ Answered by 123456 last updated on 06/Oct/16 $${y}=\mathrm{ln}\:\frac{\mathrm{5}−\mathrm{7}{x}}{\mathrm{8}+{x}} \\…
Question Number 73755 by ~blr237~ last updated on 15/Nov/19 $${show}\:{that}\:\:\:{for}\:{all}\:{integer}\:\:{n}\:,\:\:{n}+\mathrm{1}\:{divides}\:\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix} \\ $$ Answered by mind is power last updated on 15/Nov/19 $$\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}=\frac{\left(\mathrm{2}{n}\right)!}{{n}!{n}!}\in\mathbb{N} \\ $$$$\frac{{n}}{{n}+\mathrm{1}}\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}=\frac{\mathrm{2}{n}!}{{n}!.\left({n}\right)!}.\frac{{n}}{{n}+\mathrm{1}}=\frac{\mathrm{2}{n}!}{\left({n}−\mathrm{1}\right)!.\left({n}+\mathrm{1}\right)!}=\begin{pmatrix}{\mathrm{2}{n}}\\{{n}+\mathrm{1}}\end{pmatrix} \\…
Question Number 139276 by mnjuly1970 last updated on 25/Apr/21 Commented by mnjuly1970 last updated on 28/Apr/21 $$\:\:{correct}\:{answer}::\:{e}^{\mathrm{2}} .{e}^{{e}^{\mathrm{2}} } \\ $$ Answered by Dwaipayan Shikari…
Question Number 139255 by bobhans last updated on 25/Apr/21 $$ \\ $$How do you find a point on the curve y = x^2 that is…
Question Number 8113 by uchechukwu okorie favour last updated on 30/Sep/16 $${Find}\:{an}\:{equation}\:{of}\:{the}\:{tangent} \\ $$$${line}\:{to}\:{the}\:{curve}\:{y}={tan}^{\mathrm{2}} {x}\:{at}\:{the}\: \\ $$$${point}\left(\frac{{x}}{\mathrm{3}}\:,\:\mathrm{0}\right) \\ $$ Answered by prakash jain last updated…
Question Number 139171 by mnjuly1970 last updated on 23/Apr/21 Answered by Dwaipayan Shikari last updated on 23/Apr/21 $$\Gamma\left(\mathrm{1}+{x}\right)=\mathrm{1}−\gamma{x}+{O}\left({x}^{\mathrm{2}} \right)\Rightarrow\Gamma\left({x}\right)=\frac{\mathrm{1}}{{x}}−\gamma+{O}\left({x}\right) \\ $$$$\Gamma'\left(\mathrm{1}+{x}\right)=−\gamma+{O}\left({x}\right)\Rightarrow{x}\Gamma\left({x}\right)\psi\left({x}+\mathrm{1}\right)=−\gamma+{O}\left({x}\right) \\ $$$$\Rightarrow{x}\Gamma\left({x}\right)\psi\left({x}\right)+\Gamma\left({x}\right)=−\gamma+{O}\left({x}\right)\Rightarrow\psi\left({x}\right)\sim−\frac{\gamma}{{x}\Gamma\left({x}\right)}−\frac{\mathrm{1}}{{x}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}}…