Question Number 30214 by abdo imad last updated on 18/Feb/18 $${study}\:{the}\:{convergence}\:{of}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \:\:\frac{{C}_{{n}} ^{{k}} }{{k}} \\ $$$${for}\:{that}\:{use}\:{H}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:. \\ $$ Commented…
Question Number 30212 by abdo imad last updated on 18/Feb/18 $${let}\:{f}\:\:\left[{a},{b}\right]\rightarrow{R}\:{continue}\:{let}\:{suppose}\:{f}\:{derivable}\:{on}\left[{a},{b}\right] \\ $$$${and}\:\forall\:{x}\:\in\left[{a},{b}\right]\:\:{f}\left({x}\right)>\mathrm{0}\:{prove}\:{that} \\ $$$$\left.\exists{c}\in\right]{a},{b}\left[\:/\:\:\frac{{f}\left({b}\right)}{{f}\left({a}\right)}=\:{e}^{\left({b}−{a}\right)\frac{{f}^{,} \left({c}\right)}{{f}\left({c}\right)}} .\right. \\ $$ Commented by abdo imad last updated…
Question Number 30213 by abdo imad last updated on 18/Feb/18 $${p}\:{integr}\:{and}\:{p}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:\exists{c}\in\:\right]\mathrm{0},\mathrm{1}\left[\:/\right. \\ $$$${ln}\left({ln}\left({p}+\mathrm{1}\right)\right)−{ln}\left({lnp}\right)\:=\frac{\mathrm{1}}{\left({p}+{c}\right){ln}\left({p}+{c}\right)} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{ln}\left({ln}\left({p}+\mathrm{1}\right)\right)−{ln}\left({ln}\left({p}\right)\right)<\frac{\mathrm{1}}{{plnp}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\mathrm{1}}{{klnk}}=+\infty\:. \\ $$ Terms…
Question Number 30193 by abdo imad last updated on 17/Feb/18 $${let}\:{p}_{{n}} \left({x}\right)=−\mathrm{1}\:+\sum_{{k}=\mathrm{1}} ^{{k}} \:{x}^{{k}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{equation}\:{p}_{{n}} \left({x}\right)=\mathrm{0}\:{have}\:{only}\:{one}\: \\ $$$${solution}\:{x}_{{n}} \in\left[\mathrm{0},\mathrm{1}\right]\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{decreasing}\:{and}\:{minored}\:{by}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty}…
Question Number 30194 by abdo imad last updated on 17/Feb/18 $${study}\:{the}\:{convergence}\:{of}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{{C}_{{n}} ^{{k}} }\:\:{with} \\ $$$${C}_{{n}} ^{{k}} \:\:=\frac{{n}!}{{k}!\left({n}−{k}\right)!}\:. \\ $$ Commented by prof…
Question Number 30190 by abdo imad last updated on 17/Feb/18 $${study}\:{the}\:{sequence}\:{u}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}\:+{u}_{{n}} ^{\mathrm{2}} }{\mathrm{2}}}\:\:\:{with}\:−\mathrm{1}<{u}_{\mathrm{0}} <\mathrm{1}\:. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 30177 by abdo imad last updated on 17/Feb/18 $${let}\:{x}\in{R}\:\:{and}\:{u}_{{n}} =\:\prod_{{k}=\mathrm{0}} ^{{n}} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\:{find}\:{a}\:{simple}\:{form}\:{of} \\ $$$${u}_{{n}} . \\ $$ Answered by Tinkutara last updated…
Question Number 30173 by abdo imad last updated on 18/Feb/18 $${let}\:{u}_{{n}} =\:\prod_{{k}=\mathrm{1}} ^{{n}} \:\left(\mathrm{1}+\frac{{k}}{{n}^{\mathrm{2}} }\right) \\ $$$$\mathrm{1}.\:{verify}\:{that}\:{x}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{ln}\left(\mathrm{1}+{x}\right)\leqslant{x} \\ $$$$\mathrm{2}.\:{prove}\:{that}\:\left({u}_{{n}} \right)\:{is}\:{convergente}\:{and}\:{find}\:{its}\:{limit}. \\ $$ Commented by…
Question Number 30174 by abdo imad last updated on 18/Feb/18 $${let}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$$\mathrm{1}.\:{prove}\:{that}\:{ln}\left({n}+\mathrm{1}\right)\leqslant{u}_{{n}} \leqslant{ln}\left({n}\right)\:+\mathrm{1} \\ $$$$\mathrm{2}.\:{show}\:{that}\:{u}_{{n}} \:\:_{{n}\rightarrow\infty} \sim\:{ln}\left({n}\right)\:\:. \\ $$ Terms of…
Question Number 95694 by mathmax by abdo last updated on 27/May/20 $$\mathrm{solve}\:\mathrm{by}\:\mathrm{laplace}\:\mathrm{transform}\:\:\mathrm{y}^{''} \:+\mathrm{3y}^{'} +\mathrm{2y}\:=\mathrm{e}^{−\mathrm{x}} \:\:\mathrm{withy}\left(\mathrm{0}\right)=\mathrm{1}\:\mathrm{and}\:\mathrm{y}^{'} \left(\mathrm{0}\right)\:=\mathrm{2} \\ $$ Answered by mathmax by abdo last updated…