Question Number 220159 by Nicholas666 last updated on 06/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{all}\:{x},\:{y}\:\in\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:;\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{1}\:+\:{y}^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{4}\:+\:\left({x}\:+\:{y}\right)^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:+\:{y}^{\mathrm{3}} }}\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\leqslant\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} }}\:+\:\frac{\mathrm{2}}{\:^{\mathrm{4}} \sqrt{\mathrm{1}\:+\:{x}^{\mathrm{5}}…
Question Number 220121 by SdC355 last updated on 06/May/25 $$\mathrm{Let}\:{H}_{{h}} ={p}_{{h}+\mathrm{1}} /{p}_{{h}} \:,\:{p}_{{h}} \in\mathbb{P}\:,\:{p}_{\mathrm{1}} =\mathrm{2} \\ $$$$\underset{{h}=\mathrm{1}} {\overset{\infty} {\prod}}\:{H}_{{h}} =??\:\left(\underset{{h}=\mathrm{1}} {\overset{\infty} {\prod}}\:{H}_{{h}} =\frac{\mathrm{3}}{\mathrm{2}}\centerdot\frac{\mathrm{5}}{\mathrm{3}}\centerdot\frac{\mathrm{7}}{\mathrm{5}}………\right) \\ $$…
Question Number 220116 by SdC355 last updated on 06/May/25 $$\mathrm{evaluate} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{J}_{\nu} \left({z}\right){e}^{−{zt}} }{{z}^{\mathrm{2}} +\alpha^{\mathrm{2}} }\:\mathrm{d}{z}\:,\:\alpha\in\mathbb{R}\backslash\left\{\mathrm{0}\right\} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{J}_{\nu} \left({z}\right){e}^{−{zt}} }{\left({z}+\boldsymbol{{i}}\alpha\right)\left({z}−\boldsymbol{{i}}\alpha\right)}\:\mathrm{d}{z}= \\…
Question Number 220141 by MrGaster last updated on 06/May/25 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}^{\mathrm{2}} +{m}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }=? \\ $$ Answered by Nicholas666 last updated on…
Question Number 220164 by mehdee7396 last updated on 06/May/25 $${if}\:\:\alpha^{\mathrm{2}} −\mathrm{5}\alpha+\mathrm{2}=\mathrm{0}\:\:\&\:\:\beta^{\mathrm{2}} −\mathrm{5}\beta+\mathrm{2}=\mathrm{0} \\ $$$${then}\:\:\frac{\mathrm{4}\alpha+\beta^{\mathrm{5}} }{\mathrm{5}\beta^{\mathrm{2}} }=? \\ $$ Answered by Rasheed.Sindhi last updated on 07/May/25…
Question Number 220160 by Nicholas666 last updated on 06/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{all}\:{x}\:,\:{y}\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:;\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\left[\:\frac{\sqrt[{\mathrm{3}\:\:}]{\boldsymbol{{x}}^{\mathrm{3}} \:+\:\boldsymbol{{y}}^{\mathrm{3}} \:+\:\boldsymbol{\zeta}\left(\mathrm{3}\right)}}{\mathrm{1}\:+\:\boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} } \:}\:+\:\frac{\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{x}}^{\mathrm{4}} +\:\boldsymbol{\Gamma}\left(\boldsymbol{{y}}+\mathrm{1}\right)}}{\left(\mathrm{1}\:+\:\boldsymbol{{y}}^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{3}} }\:+\:\frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{5}} \:+\:\boldsymbol{{y}}^{\mathrm{5}} \right)}{\:\sqrt{\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}}…
Question Number 220131 by fantastic last updated on 06/May/25 $${If}\:\:\:{f}\left({x},{y}\right)=\frac{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{{n}} }{\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right)}+{x}\phi\left(\frac{{y}}{{x}}\right)+\Psi\left(\frac{{y}}{{x}}\right), \\ $$$${then}\:{using}\:{Euler}'{s}\:{theorem}\:{on}\:{homogenous}\:{functions},{show}\:{that} \\ $$$${x}^{\mathrm{2}} \frac{\delta^{\mathrm{2}} {f}}{\delta{x}^{\mathrm{2}} }+\mathrm{2}{xy}\frac{\delta^{\mathrm{2}} {f}}{\delta{x}\delta{y}}+{y}^{\mathrm{2}} \frac{\delta^{\mathrm{2}} {f}}{\delta{y}^{\mathrm{2}} }=\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}}…
Question Number 220094 by Nicholas666 last updated on 05/May/25 $$ \\ $$$$\:\:\:\:\mathrm{let}\:{n}\:\geqslant\:\mathrm{2}\:\in\:\mathbb{Z}\:\mathrm{and}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:…,\:{x}_{{n}} \:\mathrm{are}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{such}\:\mathrm{that}\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:{x}_{{i}} \:=\:{n}\:,\:\mathrm{prove}\:\mathrm{that}\:\:\:\: \\ $$$$\:\:\:\:\:\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{x}_{{i}} ^{{n}}…
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Question Number 220081 by SdC355 last updated on 05/May/25 Commented by MathematicalUser2357 last updated on 05/May/25 $$\boldsymbol{{Elementary}}\:\boldsymbol{{math}}\:\boldsymbol{{problem}}\:\boldsymbol{{that}}\:\mathrm{50\%}\:\boldsymbol{{of}}\:\boldsymbol{{adults}}\:\boldsymbol{{failed}} \\ $$$${Find}\:{the}\:{angle}\:{to}\:{put}\:{in}\:'?'. \\ $$ Commented by MathematicalUser2357 last…