Question Number 30488 by abdo imad last updated on 22/Feb/18 $${let}\:\:\:{a}_{{n}} =\:\prod_{{k}=\mathrm{2}} ^{{n}} \:{cos}\left(\frac{\pi}{\mathrm{2}^{{k}} }\right)\:.{prove}\:{that}\:\left({a}_{{n}} \right)\:{ks}\:{decreasing}. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{b}_{{n}} ={a}_{{n}} {cos}\left(\frac{\pi}{\mathrm{2}^{{n}} }\right)\:\:{find}\:{lim}_{{n}\rightarrow\infty} \left({a}_{{n}} \:−{b}_{{n}} \right). \\…
Question Number 30486 by abdo imad last updated on 22/Feb/18 $${let}\:{give}\:{a}_{{n}} =\:\prod_{{k}=\mathrm{1}} ^{{n}} \:{cos}\left(\:\frac{\pi}{\left({k}+\mathrm{2}\right)!}\right)\:{and} \\ $$$${b}_{{n}} =\prod_{{k}=\mathrm{1}} ^{{n}} \:\:{sin}\left(\frac{\pi}{\left({k}+\mathrm{2}\right)!}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{foe}\:{a}_{{n}} \:{and}\:{b}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{a}_{{n}} .{b}_{{n}}…
Question Number 30485 by abdo imad last updated on 22/Feb/18 $$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{x}>\mathrm{0}\:\:\:\frac{{x}}{{x}+\mathrm{1}}\:\leqslant\:{ln}\left(\mathrm{1}+{x}\right)\leqslant{x} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{put}\:\:{S}_{{n}} =\:\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}\:+\:\frac{{k}}{{n}}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{S}_{{n}} \:\:. \\ $$ Commented by chantriachheang last updated…
Question Number 30484 by abdo imad last updated on 22/Feb/18 $$\left.\mathrm{1}\right)\:{prove}\:{that}\:{if}\:{f}\:{is}\:{decreasing}\:{function}\:{we}\:{have} \\ $$$$\:\int_{{n}} ^{{n}+\mathrm{1}} {f}\left({t}\right){dt}\:<{f}\left({n}\right)<\:\int_{{n}−\mathrm{1}} ^{{n}} \:{f}\left({t}\right)\:{dt}\:\:. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{put}\:\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}^{\mathrm{2}} } \:\:\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{{k}}}\:.{calculate}\:\left[{S}_{{n}} \right]. \\…
Question Number 30483 by abdo imad last updated on 22/Feb/18 $${we}\:{define}\:{the}\:{bernoulli}\:{polynomial}\:{B}_{{n}} \:{by} \\ $$$${b}_{\mathrm{0}} =\mathrm{1}\:{and}\:\forall{n}\in\:{N}^{\bigstar} \:\:\:{b}_{{n}} ^{'} ={n}\:{b}_{{n}−\mathrm{1}} \:\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {b}_{{n}} \left({t}\right){dt}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{b}_{{n}} \left(\mathrm{1}\right)−{b}_{{n}}…
Question Number 30443 by abdo imad last updated on 22/Feb/18 $${let}\:\:{w}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{x}^{{k}} }{{k}}\:\:{find}\:{w}_{{n}} \left({x}\right)\:{for}\:\mid{x}\mid<\mathrm{1}\: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} {w}_{{n}} \left({x}\right). \\ $$ Terms of Service…
Question Number 30438 by abdo imad last updated on 22/Feb/18 $${let}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:{prove}\:{that}\: \\ $$$$\xi\left({x}\right)=\:\gamma\:+\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+{o}\left(\mathrm{1}\right). \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 30439 by abdo imad last updated on 22/Feb/18 $${let}\:{F}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{{x}} }\:\:{calculate}\:\frac{{dF}}{{dx}}\left({x}\right). \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 30440 by abdo imad last updated on 22/Feb/18 $${if}\:\:\:\left(\sum_{{n}\geqslant\mathrm{1}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{{x}} }\right)^{\mathrm{2}} =\:\sum_{{n}} \:\:{c}_{{n}\:} \left({x}\right)\:\:{find}\:{c}_{{n}} \left({x}\right). \\ $$ Terms of Service Privacy Policy…
Question Number 30436 by abdo imad last updated on 22/Feb/18 $${let}\:\varphi\left({x}\right)=\mathrm{1}−\mathrm{2}^{\mathrm{1}−{x}} \:\:{prove}\:{that} \\ $$$$\varphi\left({x}\right)=\left({x}−\mathrm{1}\right){ln}\mathrm{2}\:−\frac{\left({ln}\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{2}}\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:+{o}\left(\left({x}−\mathrm{1}\right)^{\mathrm{2}} \right). \\ $$ Commented by prof Abdo imad last…