Question Number 1635 by 123456 last updated on 28/Aug/15 $$\mathrm{lets}\:{x}>\mathrm{0},\:\mathrm{and}\:\mathrm{take}\:\mathrm{the}\:\mathrm{sequence}\:{a} \\ $$$${a}_{\mathrm{0}} =\sqrt{{x}} \\ $$$${a}_{{n}+\mathrm{1}} =\sqrt{{x}+{a}_{{n}} } \\ $$$$\mathrm{i}.\mathrm{proof}\:\mathrm{that}\:\mathrm{0}\leqslant{a}_{{n}} \leqslant{a}_{{n}+\mathrm{1}} \\ $$$$\mathrm{ii}.\mathrm{proof}\:\mathrm{that}\:\exists\mathrm{M}\:\mathrm{such}\:\mathrm{that}\:{a}_{{n}} \leqslant\mathrm{M} \\ $$$$\mathrm{iii}.\mathrm{using}\:\mathrm{i}\:\mathrm{and}\:\mathrm{ii}\:\mathrm{proof}\:\mathrm{that}\:\underset{{n}\rightarrow\infty}…
Question Number 67055 by Tony Lin last updated on 22/Aug/19 $${let}\:\mathbb{Z}_{+} =\mathbb{N}\cup\left\{\mathrm{0}\right\},\:{f}:\:\mathbb{Z}_{+} ×\mathbb{Z}_{+} \rightarrow\mathbb{Z}_{+} \\ $$$${f}\left({m},\:{n}\right)=\frac{\left({m}+{n}\right)\left({m}+{n}+\mathrm{1}\right)}{\mathrm{2}}+{m} \\ $$$${prove}\:{that}\:{f}\:{is}\:{a}\:{one}-{to}-{one}\:{function} \\ $$$${and}\:{also}\:{an}\:{onto}\:{function} \\ $$ Terms of Service…
Question Number 67057 by Tony Lin last updated on 22/Aug/19 $$\left(\mathrm{1}\right){find}\:\cap_{{n}=\mathrm{1}} ^{\infty} \left[\mathrm{0},\:\frac{\mathrm{1}}{{n}}\right) \\ $$$$\left(\mathrm{2}\right){find}\:\cup_{{n}=\mathrm{2}} ^{\infty} \left[\frac{\mathrm{1}}{{n}},\:\mathrm{1}−\frac{\mathrm{1}}{{n}}\right] \\ $$ Answered by Kunal12588 last updated on…
Question Number 67034 by mathmax by abdo last updated on 22/Aug/19 $${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{cos}\left(\mathrm{2}{nx}\right)}{{n}} \\ $$ Commented by mathmax by abdo last updated on 23/Aug/19…
Question Number 132574 by liberty last updated on 15/Feb/21 $$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{log}\:_{\mathrm{2020}} \left(\mathrm{x}\right)\:\mathrm{and}\: \\ $$$$\mathrm{p}^{\left(\mathrm{p}\right)^{\mathrm{p}^{\mathrm{2020}} } } \:=\:\mathrm{2020}\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{f}\left(\mathrm{p}\right)\:=\:… \\ $$$$\left(\mathrm{a}\right)\sqrt[{\mathrm{2020}}]{\mathrm{2020}}\:\:\:\:\:\left(\mathrm{c}\right)\:\sqrt[{\mathrm{2020}}]{\frac{\mathrm{1}}{\mathrm{2020}}} \\ $$$$\left(\mathrm{b}\right)\:\frac{\mathrm{1}}{\mathrm{2020}}\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{2020}\:\:\:\:\:\:\left(\mathrm{e}\right)\:\mathrm{log}\:_{\mathrm{10}} \left(\mathrm{2020}\right) \\ $$…
Question Number 67023 by mathmax by abdo last updated on 21/Aug/19 $${find}\:{the}\:{sequence}\:{U}_{{n}} \:{wich}\:{verify}\:\:{U}_{{n}} +{U}_{{n}+\mathrm{1}} ={sin}\left({n}\right)\:\:\forall{n}\:{from}\:{n} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 67014 by mathmax by abdo last updated on 21/Aug/19 $${solve}\:{y}^{''} +{x}^{\mathrm{2}} {y}^{'} ={e}^{−{x}} {sin}\left(\mathrm{3}{x}\right) \\ $$ Commented by mathmax by abdo last updated…
Question Number 67015 by mathmax by abdo last updated on 21/Aug/19 $${let}\:{f}\left({x}\right)\:={arctan}\left(\mathrm{1}+{e}^{−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \right) \\ $$$${calculate}\:{f}^{'} \left({x}\right)\:\:{and}\:{f}^{''} \left({x}\right). \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow−\infty} \:\:\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{variation}\:{of}\:{f}\left({x}\right) \\…
Question Number 67010 by mathmax by abdo last updated on 21/Aug/19 $${calculate}\:\:\sum_{{n}=\mathrm{4}} ^{+\infty} \:\:\:\:\frac{{n}}{\left({n}^{\mathrm{2}} −\mathrm{9}\right)^{\mathrm{2}} } \\ $$ Commented by mathmax by abdo last updated…
Question Number 67013 by mathmax by abdo last updated on 21/Aug/19 $${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right){n}^{\mathrm{3}} } \\ $$ Commented by mathmax by abdo last updated…