Question Number 40882 by prof Abdo imad last updated on 28/Jul/18 $$\left.\mathrm{1}\right){prove}\:{that}\:\forall{n}\geqslant\mathrm{2}\left({n}\:{inyegr}\right) \\ $$$${x}^{\mathrm{2}{n}} −\mathrm{1}=\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \left({x}^{\mathrm{2}} \:−\mathrm{2}{cos}\left(\frac{{k}\pi}{{n}}\right){x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcost}\:+\mathrm{1}\right){dt} \\ $$…
Question Number 40880 by prof Abdo imad last updated on 28/Jul/18 $${prove}\:{that}\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:\:\frac{\mathrm{1}}{\left(\alpha−\mathrm{1}\right){n}^{\alpha−\mathrm{1}} }{with}\:\alpha>\mathrm{1} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 40876 by prof Abdo imad last updated on 28/Jul/18 $${prove}\:{by}\:{recurrence}\:{that}\: \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{4}} =\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{3}{n}−\mathrm{1}\right)}{\mathrm{30}} \\ $$ Terms of Service Privacy Policy…
Question Number 40878 by prof Abdo imad last updated on 28/Jul/18 $${let}\:{u}_{\mathrm{0}} >\mathrm{0}\:{and}\:\forall{n}\in{N} \\ $$$${u}_{{n}+\mathrm{1}} ={u}_{{n}} \:+\frac{\mathrm{1}}{{u}_{{n}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{increasing}\:{and}\:{lim}\:{u}_{{n}} \:=+\infty \\ $$$$\left.\mathrm{2}\right){by}\:{consideringthe}\:{function}\varphi\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}{t}+{x}} \\…
Question Number 106408 by bemath last updated on 05/Aug/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{k}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{4}\:\mathrm{intersect} \\ $$$$\mathrm{the}\:\mathrm{parabola}\:\mathrm{y}=\:\mathrm{x}^{\mathrm{2}} +\mathrm{k}\: \\ $$ Answered by 1549442205PVT last updated on 06/Aug/20 $$\mathrm{The}\:\mathrm{circle}\:\mathrm{intersection}\:\mathrm{the}\:\mathrm{parabol}…
Question Number 106315 by pticantor last updated on 04/Aug/20 $$\boldsymbol{{show}}\:\boldsymbol{{that}} \\ $$$$\circledast\forall\:\left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,…..,\boldsymbol{{x}}_{\boldsymbol{{n}}\:} \right)\in\mathbb{R}^{\boldsymbol{{n}}} \\ $$$$\left(\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\boldsymbol{{x}}_{\boldsymbol{{k}}} \right)^{\mathrm{2}} \leqslant\boldsymbol{{n}}\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\boldsymbol{{x}}_{\boldsymbol{{k}}} ^{\mathrm{2}} \\…
Question Number 106309 by bemath last updated on 04/Aug/20 $$\mathrm{If}\:\mathrm{g}\left(\mathrm{x}\right)=\:\mathrm{x}+\sqrt{\mathrm{x}}\:\mathrm{and}\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{2}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{ax}+\mathrm{b}}=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left(\mathrm{f}\circ\mathrm{g}\right)'\left(\mathrm{1}\right). \\ $$ Answered by bobhans last updated on 05/Aug/20 $$\mathrm{g}\left(\mathrm{x}\right)=\:\mathrm{x}+\sqrt{\mathrm{x}}\:\Rightarrow\mathrm{g}'\left(\mathrm{x}\right)=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{x}}}\:;\:\mathrm{g}'\left(\mathrm{1}\right)=\:\frac{\mathrm{3}}{\mathrm{2}} \\…
Question Number 106138 by bobhans last updated on 03/Aug/20 $$\mathrm{If}\:\mathrm{root}\:\mathrm{of}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{3}} −\mathrm{px}^{\mathrm{2}} +\mathrm{qx}−\mathrm{r}=\mathrm{0}\:\mathrm{are}\:\mathrm{in} \\ $$$$\mathrm{AP}\:\mathrm{than}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{p},\mathrm{q} \\ $$$$\mathrm{and}\:\mathrm{r}\:? \\ $$ Answered by bemath last updated on 03/Aug/20…
Question Number 106133 by mathmax by abdo last updated on 02/Aug/20 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{e}^{−\mathrm{2x}} \mathrm{ln}\left(\mathrm{3}−\mathrm{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$ Answered by mathmax…
Question Number 106134 by mathmax by abdo last updated on 03/Aug/20 $$\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)\:=\mathrm{arcatan}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}−\mathrm{2x}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{find}\:\mathrm{g}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{g}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$$$\mathrm{3}/\:\mathrm{calculate}\:\:\int_{−\frac{\mathrm{1}}{\mathrm{4}}} ^{\frac{\mathrm{1}}{\mathrm{4}}} \:\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx} \\ $$ Terms…