Question Number 2936 by Rasheed Soomro last updated on 30/Nov/15 $${Suppose}\:{I}\:{want}\:{to}\:{prove}\:{a}\:{statement},{say}\:{P}\left({n}\right),\: \\ $$$${for}\:{natural}\:{numbers}\:\boldsymbol{{mathematical}}\:\boldsymbol{{induction}} \\ $$$${is}\:{a}\:{proper}\:{tool}\:{for}\:{me}. \\ $$$$\mathcal{I}{f}\:{a}\:{P}\left({n}\right)\:{is}\:{required}\:{to}\:{prove}\:{for}\:{natural}\:{n}\geqslant{c}\: \\ $$$${mathematical}\:{induction}\:{is}\:{again}\:{a}\:{tool}\:{of}\:{proof}. \\ $$$$ \\ $$$$\mathcal{N}{ow}\:{suppose}\:{I}\:{have}\:{a}\:{statement}\:{which}\:{is}\:{true} \\ $$$${only}\:{for}\:\:\underset{−}…
Question Number 134005 by mr W last updated on 26/Feb/21 $${solve}\:{x}^{\mathrm{3}} −\mathrm{2}\lfloor{x}\rfloor=\mathrm{5} \\ $$ Answered by MJS_new last updated on 26/Feb/21 $${x}={i}\left[\mathrm{nteger}\:\mathrm{part}\right]+{f}\left[\mathrm{ractal}\:\mathrm{part}\right] \\ $$$$\left({i}+{f}\right)^{\mathrm{3}} −\mathrm{2}{i}=\mathrm{5}…
Question Number 68470 by mathmax by abdo last updated on 11/Sep/19 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$ Commented by mathmax by abdo last updated…
Question Number 134006 by mohammad17 last updated on 26/Feb/21 Answered by malwan last updated on 26/Feb/21 $$\left(\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{4}} {{lim}}\:\frac{\mathrm{2}{x}−\mathrm{5}}{\mathrm{3}}\:=\:\frac{\mathrm{2}×\mathrm{4}−\mathrm{5}}{\mathrm{3}}\:=\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow−\mathrm{3}^{+} } {{lim}}\:\frac{\sqrt{{x}+\mathrm{4}}\:−\sqrt{−{x}−\mathrm{2}}}{\:\sqrt{{x}+\mathrm{3}}}×\frac{\sqrt{{x}+\mathrm{4}}\:+\:\sqrt{−{x}−\mathrm{2}}}{\:\sqrt{{x}+\mathrm{4}}\:+\:\sqrt{−{x}−\mathrm{2}}} \\ $$$$=\:\underset{{x}\rightarrow−\mathrm{3}^{+} }…
Question Number 2932 by Filup last updated on 30/Nov/15 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{subfactorial}\:\mathrm{function}? \\ $$$${i}.{e}.\:\:\:\:\:!{x} \\ $$ Commented by 123456 last updated on 30/Nov/15 $$\mathrm{number}\:\mathrm{of}\:\mathrm{dessaragment} \\ $$ Commented…
Question Number 68466 by mathmax by abdo last updated on 11/Sep/19 $${let}\:{f}\left({x}\right)\:=\int_{{x}^{} } ^{{x}^{\mathrm{2}} −{x}} \:{arctan}\left({e}^{−{x}−{t}} \right){dt} \\ $$$${calculate}\:{f}^{'} \left({x}\right)\:\:\:{and}\:{f}^{'} \left(\mathrm{0}\right). \\ $$ Commented by…
Question Number 2931 by Filup last updated on 30/Nov/15 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{double}\:\mathrm{factorial}\:\mathrm{function}? \\ $$$${i}.{e}.\:\:\:\:\:{x}!! \\ $$ Answered by 123456 last updated on 30/Nov/15 $${x}!!={x}\left({x}−\mathrm{2}\right)!!,{x}\in\mathbb{N} \\ $$$$\mathrm{0}!!=\mathrm{1} \\…
Question Number 2930 by Filup last updated on 30/Nov/15 $$\mathrm{Prove}\:\mathrm{that}\:\Gamma\left({i}\right)=−{i}\left({i}\right)! \\ $$$$\mathrm{where}\:{i}=\sqrt{−\mathrm{1}} \\ $$ Commented by 123456 last updated on 02/Dec/15 $$\Gamma\left({z}\right)\Gamma\left(\mathrm{1}−{z}\right)=\frac{\pi}{\mathrm{sin}\:\pi{z}} \\ $$$$\Gamma\left({z}\right)\Gamma\left(−{z}\right)=−\frac{\pi}{{z}\:\mathrm{sin}\:\pi{z}} \\…
Question Number 134002 by Dwaipayan Shikari last updated on 26/Feb/21 $${What}\:{will}\:{be}\:{the}\:{minimum}\:{area}\:{of}\:{a}\:{heptagon}\:{inscribed}\:{in} \\ $$$${an}\:{unit}\:{square}? \\ $$ Commented by Dwaipayan Shikari last updated on 26/Feb/21 Answered by…
Question Number 133997 by rs4089 last updated on 26/Feb/21 $${lim}_{{n}\rightarrow\infty} \left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…+\frac{\mathrm{1}}{{n}}}{{n}^{\mathrm{2}} }\right)^{{n}} \\ $$ Answered by mathmax by abdo last updated on 27/Feb/21 $$\mathrm{U}_{\mathrm{n}} =\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+….+\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{n}^{\mathrm{2}}…