Question Number 88364 by Power last updated on 10/Apr/20 Answered by mind is power last updated on 10/Apr/20 $${by}\:{Cauchy}\:{shwart} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{1}.{x}_{{i}} \leqslant\sqrt{\underset{{i}=\mathrm{1}} {\overset{{n}}…
Question Number 153897 by mathdanisur last updated on 11/Sep/21 $$\mathrm{Denote}\:\:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \:\:\mathrm{is}\:\mathrm{the}\:\mathrm{unique}\:\mathrm{positive}\:\mathrm{root} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{x}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \:+\:…\:\mathrm{x}\:=\:\mathrm{n}\:+\:\mathrm{2} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left(\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)\:\mathrm{converges} \\ $$$$\mathrm{to}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{Find}\:\mathrm{that} \\ $$$$\mathrm{limit}. \\…
Question Number 153896 by SANOGO last updated on 11/Sep/21 Answered by ARUNG_Brandon_MBU last updated on 12/Sep/21 $$\Omega=\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{ln}{x}}{\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx},\:{x}=\frac{\mathrm{1}}{{u}}\Rightarrow{dx}=−\frac{\mathrm{1}}{{u}^{\mathrm{2}} }{du} \\ $$$$\:\:\:\:=−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{u}\mathrm{ln}{u}}{\left({u}+\mathrm{1}\right)\left({u}^{\mathrm{2}}…
Question Number 153899 by mathdanisur last updated on 11/Sep/21 $$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{there}\:\mathrm{exists}\:\:\mathrm{2016} \\ $$$$\mathrm{distinct}\:\mathrm{prime}\:\mathrm{numbers}\:\:\mathrm{p}_{\mathrm{1}} ,\mathrm{p}_{\mathrm{2}} ,…,\mathrm{p}_{\mathrm{2016}} \\ $$$$\mathrm{and}\:\mathrm{positive}\:\mathrm{integer}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\mathrm{2016}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{p}_{\boldsymbol{\mathrm{i}}} ^{\mathrm{2}} \:+\:\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} } \\ $$…
Question Number 88360 by jagoll last updated on 10/Apr/20 $$\frac{\mathrm{e}}{\:\sqrt{\mathrm{e}}}\:×\:\frac{\sqrt[{\mathrm{3}\:\:}]{\mathrm{e}}}{\:\sqrt[{\mathrm{4}\:\:}]{\mathrm{e}}}\:×\:\frac{\sqrt[{\mathrm{5}\:\:}]{\mathrm{e}}}{\:\sqrt[{\mathrm{6}\:\:}]{\mathrm{e}}}\:×\:\frac{\sqrt[{\mathrm{7}\:\:}]{\mathrm{e}}}{\:\sqrt[{\mathrm{8}\:\:}]{\mathrm{e}}}×…=? \\ $$ Commented by john santu last updated on 10/Apr/20 $$=\:{e}^{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{6}}+…} \\ $$$$\left[\:\mathrm{ln}\:\left(\mathrm{1}+\mathrm{x}\right)\:=\:\mathrm{x}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{3}}−\frac{\mathrm{x}^{\mathrm{4}}…
Question Number 153898 by mathdanisur last updated on 11/Sep/21 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{functions}\:\:\mathrm{f}:\mathrm{Q}\rightarrow\mathrm{Q}\:\:\mathrm{satisfying} \\ $$$$\mathrm{these}\:\mathrm{followong}\:\mathrm{conditions}\:\mathrm{for}\:\mathrm{all}\:\boldsymbol{\mathrm{x}}\in\mathrm{Q} \\ $$$$\mathrm{1}.\:\mathrm{f}\left(\mathrm{x}\:+\:\mathrm{1}\right)\:=\:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{1} \\ $$$$\mathrm{2}.\:\mathrm{f}\left(\mathrm{x}^{\mathrm{3}} \right)\:=\:\mathrm{f}^{\:\mathrm{3}} \left(\mathrm{x}\right) \\ $$ Answered by talminator2856791 last updated…
Question Number 153893 by Tawa11 last updated on 11/Sep/21 $$\int_{\:\mathrm{0}} ^{\:\:\infty} \mathrm{a}\:\underset{\mathrm{p}\:\rightarrow\:\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{p}^{\mathrm{2}} \:\:−\:\:\:\mathrm{x}^{\mathrm{2n}} }{\mathrm{p}^{\mathrm{2}} }\right)\mathrm{dx},\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:<\:\:\mathrm{2n}\:\:<\:\:\mathrm{n}\:\:+\:\:\mathrm{1} \\ $$ Terms of Service Privacy Policy Contact:…
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Question Number 88357 by jagoll last updated on 10/Apr/20 $$\int\underset{\mathrm{1}} {\overset{\mathrm{4}} {\:}}\:\frac{\mathrm{dx}}{\left(\mathrm{4x}−\mathrm{1}\right)\sqrt{\mathrm{x}}} \\ $$ Commented by john santu last updated on 10/Apr/20 $$\left[\:{t}=\sqrt{{x}}\:,\:\mathrm{2}{t}\:{dt}\:=\:{dx}\:\right]\: \\ $$$$\underset{\mathrm{1}}…
Question Number 22820 by Physics lover last updated on 22/Oct/17 $${total}\:{number}\:{of}\:{permutations}\:{of} \\ $$$${five}\:{abjects}\:\rightarrow\:{A},{A},{A},{B},{B}\: \\ $$$${in}\:{a}\:{circle}? \\ $$ Commented by ajfour last updated on 22/Oct/17 $${four}\:.…