Question Number 150059 by mathdanisur last updated on 09/Aug/21 $$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}\centerdot\left(\mathrm{2n}\:+\:\mathrm{1}\right)}\:=\:? \\ $$ Answered by Kamel last updated on 09/Aug/21 $${S}=\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{2}{n}}−\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\right)=\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{+\infty}…
Question Number 150058 by mathdanisur last updated on 09/Aug/21 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{cos}^{\mathrm{4}} \left(\mathrm{x}\right)\:+\:\mathrm{i}\:\mathrm{sin}^{\mathrm{4}} \left(\mathrm{x}\right)\:=\:\mathrm{4e}^{\mathrm{4}\boldsymbol{\mathrm{ix}}} \\ $$ Commented by MJS_new last updated on 10/Aug/21 $$\mathrm{I}\:\mathrm{found}\:\mathrm{these}\:\left({n}\in\mathbb{Z}\right): \\…
Question Number 150053 by mathdanisur last updated on 09/Aug/21 $$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{t}=\mathrm{4} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{4}}{\left(\mathrm{xyzt}\right)^{\mathrm{2}} }\:+\:\mathrm{3}\:\geqslant\:\sqrt{\mathrm{45}\:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{t}^{\mathrm{4}} } \\ $$ Terms of Service Privacy…
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Question Number 84515 by M±th+et£s last updated on 13/Mar/20 $${Q}.{solve} \\ $$$${x}^{\mathrm{3}} −{x}={x}! \\ $$ Answered by mr W last updated on 13/Mar/20 $$\left({x}+\mathrm{1}\right){x}\left({x}−\mathrm{1}\right)={x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)! \\…
Question Number 84512 by 698148290 last updated on 13/Mar/20 Answered by jagoll last updated on 14/Mar/20 $$\mathrm{tangent}\:\mathrm{equation}\:\mathrm{at}\:\mathrm{point} \\ $$$$\mathrm{P}\left(\mathrm{2a}+\mathrm{2t}\:,\:\frac{\mathrm{at}^{\mathrm{2}} }{\mathrm{2}}\right)\:\mathrm{to}\:\mathrm{the}\:\mathrm{parabola} \\ $$$$\left(\mathrm{x}−\mathrm{2a}\right)^{\mathrm{2}} \:=\:\mathrm{2ay}\: \\ $$$$\Rightarrow\:\mathrm{2t}\left(\mathrm{x}−\mathrm{2a}\right)\:=\:\mathrm{ay}\:+\:\frac{\left(\mathrm{at}\right)^{\mathrm{2}}…
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Question Number 84510 by 698148290 last updated on 13/Mar/20 Answered by jagoll last updated on 14/Mar/20 $$\mathrm{equation}\:\mathrm{of}\:\mathrm{tangent} \\ $$$$\Rightarrow\:\frac{\mathrm{x}_{\mathrm{1}} \mathrm{x}}{\mathrm{a}^{\mathrm{2}} }\:+\:\frac{\mathrm{y}_{\mathrm{1}} \mathrm{y}}{\mathrm{b}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$$\Rightarrow\:\frac{\mathrm{cos}\:\theta\:\mathrm{x}}{\mathrm{a}}\:+\:\frac{\mathrm{sin}\:\theta\:\mathrm{y}}{\mathrm{b}}\:=\:\mathrm{1}\:…
Question Number 150044 by bramlexs22 last updated on 09/Aug/21 $$\:\mathrm{Given}\:\mathrm{f}\left(\frac{\mathrm{2x}−\mathrm{3}}{\mathrm{2x}+\mathrm{1}}\right)+\mathrm{f}\left(\frac{\mathrm{2x}+\mathrm{3}}{\mathrm{1}−\mathrm{2x}}\right)=\:\mathrm{4x} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$ Answered by liberty last updated on 09/Aug/21 $$\:\mathrm{f}\left(\frac{\mathrm{2x}−\mathrm{3}}{\mathrm{2x}+\mathrm{1}}\right)+\mathrm{f}\left(\frac{\mathrm{2x}+\mathrm{3}}{\mathrm{1}−\mathrm{2x}}\right)=\mathrm{4x}\:\ldots\left(\mathrm{i}\right) \\ $$$$\mathrm{let}\:\frac{\mathrm{2x}−\mathrm{3}}{\mathrm{2x}+\mathrm{1}}\:=\:\mathrm{y}\:; \\…
Question Number 150040 by bobhans last updated on 09/Aug/21 $$\mathrm{Let}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1}\:.\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\:\frac{\mathrm{ab}}{\mathrm{1}−\mathrm{c}^{\mathrm{2}} }\:+\frac{\mathrm{bc}}{\mathrm{1}−\mathrm{a}^{\mathrm{2}} }+\frac{\mathrm{ca}}{\mathrm{1}−\mathrm{b}^{\mathrm{2}} }\:\leqslant\:\frac{\mathrm{3}}{\mathrm{8}} \\ $$ Terms of Service Privacy Policy Contact:…