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Question Number 191787 by Mastermind last updated on 30/Apr/23 $$\mathrm{Ques}.\:\mathrm{2}\:\left(\mathrm{Metric}\:\mathrm{Space}\:\mathrm{Question}\right) \\ $$$$\:\:\:\:\:\:\mathrm{Let}\:\mathrm{d}\:\mathrm{be}\:\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\mathrm{a}\:\mathrm{non}−\mathrm{empty} \\ $$$$\mathrm{set}\:\mathrm{X}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:\mathrm{U}\:\mathrm{is} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\mathrm{U}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{d}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{1}+\mathrm{d}\left(\mathrm{x},\mathrm{y}\right)},\:\mathrm{where} \\ $$$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{arbitrary}\:\mathrm{element}\:\mathrm{X}\:\mathrm{is}\:\mathrm{also} \\ $$$$\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\mathrm{X}. \\ $$ Terms of Service…
Question Number 191786 by Mastermind last updated on 30/Apr/23 $$\mathrm{Ques}.\:\mathrm{1}\:\left(\mathrm{Metric}\:\mathrm{Space}\:\mathrm{Question}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Let}\:\mathrm{X}\:=\:\rho_{\infty} \:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\: \\ $$$$\mathrm{bounded}\:\mathrm{sequences}\:\mathrm{of}\:\mathrm{complex}\: \\ $$$$\mathrm{numbers}.\:\mathrm{That}\:\mathrm{is}\:\mathrm{every}\:\mathrm{element}\:\mathrm{of} \\ $$$$\rho_{\infty} \:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{sequence}\:\overset{−} {\mathrm{x}}=\left\{\overset{−} {\mathrm{x}}\right\}_{\mathrm{k}=\mathrm{1}} ^{\infty} \: \\…
Question Number 60706 by maxmathsup by imad last updated on 24/May/19 $${let}\:{f}\left({x}\right)\:=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{let}\:{U}_{{n}} ={f}^{\left({n}\right)} \left({x}\right)\:\:{prove}\:{that}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{U}_{{k}} {U}_{{n}+\mathrm{1}−{k}} =\mathrm{0}\:\:{for}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:.…
Question Number 60705 by behi83417@gmail.com last updated on 24/May/19 Answered by MJS last updated on 25/May/19 $$\mathrm{the}\:\mathrm{only}\:\mathrm{solution}\:\mathrm{is}\:{x}={y}=\mathrm{1} \\ $$ Commented by behi83417@gmail.com last updated on…
Question Number 126238 by mohammad17 last updated on 18/Dec/20 Answered by Olaf last updated on 18/Dec/20 $$\Omega\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{{y}^{\mathrm{2}} } ^{\mathrm{1}} \frac{{e}^{{x}} }{\:\sqrt{{x}}}{dxdy} \\ $$$$\mathrm{Let}\:{u}\:=\:\sqrt{{x}}…
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Question Number 191775 by Mingma last updated on 30/Apr/23 Answered by AST last updated on 30/Apr/23 $$\overset{\_\_\_\_\_\_\_\_\_\_\_\_} {{ABCDEF}}=\mathrm{10000}\overset{\_\_\_\_\_\_\_} {{ABCD}}+\overset{\_\_\_} {{EF}} \\ $$$$\mathrm{17}\left[\mathrm{10000}\overset{\_\_\_} {{AB}}+\overset{\_\_\_\_\_\_\_} {{CDEF}}\right]=\mathrm{12}\left[\mathrm{100}\overset{\_\_\_\_\_\_\_} {{CDEF}}+\overset{\_\_\_}…
Question Number 60701 by maxmathsup by imad last updated on 24/May/19 $${let}\:{f}\left({x}\right)\:={e}^{−{x}^{\mathrm{2}} } {ln}\left(\mathrm{2}−{x}\right)\:\:\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$ Commented by maxmathsup by imad last updated on 27/May/19…
Question Number 60697 by ajfour last updated on 24/May/19 Commented by ajfour last updated on 24/May/19 $${How}\:{far}\:{can}\:{the}\:{slider}\:{slide}\:{if}\:{it} \\ $$$${ejects}\:{mass}\:{at}\:{rate}\:\frac{{dM}}{{dt}}\:=\:{R}\:,\:{at}\:{a} \\ $$$${relative}\:{speed}\:{u}\:?\: \\ $$$$\left(\frac{{M}_{\mathrm{0}} }{\mathrm{2}}\:\:{of}\:{M}_{\mathrm{0}} \:{doesn}'{t}\:{burn}\:{and}\right.…