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Question Number 162100 Answers: 1 Comments: 1
Question Number 162099 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{PROVE}\:\:\:\mathscr{THAT}\:\: \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{Li}_{\:\mathrm{2}} \:\left({x}\:\right).\:\mathrm{ln}\left(\:{x}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:−\pi^{\:\mathrm{4}} }{\mathrm{90}} \\ $$$$\:\:\:\:\:−−−−−−−−−− \\ $$$$\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\:\left({x}\:\right)\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{x}^{\:{n}−\mathrm{1}} }{{n}^{\:\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{x}^{\:{n}−\mathrm{1}} .\:\mathrm{ln}\left({x}\:\right)\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\left\{\left[\:\frac{{x}^{\:{n}} }{{n}}\:\mathrm{ln}\left(\:{x}\:\right)\right]_{\mathrm{0}} ^{\:\mathrm{1}} −\frac{\mathrm{1}}{{n}}\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\:{n}−\mathrm{1}} {dx}\right\} \\ $$$$\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{−\mathrm{1}}{{n}^{\:\mathrm{4}} }\:=\:−\:\zeta\:\left(\mathrm{4}\:\right)\:=\:\frac{−\pi^{\:\mathrm{4}} }{\:\mathrm{90}}\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:−−−\:\mathscr{M}\:.\:\mathscr{N}\:\:−−−\: \\ $$$$ \\ $$
Question Number 162139 Answers: 2 Comments: 0
Question Number 162138 Answers: 1 Comments: 0
Question Number 162118 Answers: 0 Comments: 7
$${determiner}\:{le}\:{reste}\:{de}\:{la}\:{division}\:{euclidienne}\:{de} \\ $$$$\mathrm{10}^{\mathrm{100}} \:{par}\:\mathrm{105} \\ $$
Question Number 162117 Answers: 2 Comments: 1
Question Number 162092 Answers: 0 Comments: 0
Question Number 162088 Answers: 2 Comments: 0
$${x}^{\mathrm{2}} =\mathrm{2}^{{x}} \\ $$$${solve}\:\:\:{for}\:\:\:\:{x}=? \\ $$
Question Number 162083 Answers: 0 Comments: 0
Question Number 162081 Answers: 2 Comments: 0
$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:−\sqrt{\frac{\mathrm{1}}{{x}}−\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:=?\right. \\ $$
Question Number 162165 Answers: 2 Comments: 1
$$\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$$\:{ax}^{\mathrm{2}} +\mathrm{2}{bx}+{c}=\mathrm{0}\:\mathrm{is}\:\mathrm{same}\:\mathrm{as}\:\mathrm{the}\:\mathrm{ratio} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{px}^{\mathrm{2}} +\mathrm{2}{qx}+{r}=\mathrm{0}\:\mathrm{where} \\ $$$$\:{a},{b},{c},{p}\:,{r}\:\mathrm{are}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left(\frac{{b}^{\mathrm{2}} }{{q}^{\mathrm{2}} }\right)\left(\frac{{p}}{{a}}\right)\left(\frac{{r}}{{c}}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\: \\ $$
Question Number 162074 Answers: 1 Comments: 0
$${Find}\:\:{the}\:\:{exact}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1004}} {\sum}}\:\left(\underset{{k}} {\overset{\mathrm{2014}} {\:}}\right)\:\centerdot\:\mathrm{3}^{{k}} \:. \\ $$
Question Number 162071 Answers: 1 Comments: 9
Question Number 162068 Answers: 1 Comments: 0
Question Number 162066 Answers: 0 Comments: 0
$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }=??? \\ $$
Question Number 162062 Answers: 1 Comments: 0
$$\Omega\left(\alpha;\beta\right)\:=\underset{\:-\mathrm{1}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}\boldsymbol{\alpha}-\mathrm{1}} \:\left(\mathrm{1}-\mathrm{x}\right)^{\mathrm{2}\boldsymbol{\beta}-\mathrm{1}} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\boldsymbol{\alpha}+\boldsymbol{\beta}} }\:\mathrm{dx}\:;\:\alpha;\beta>\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{form}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\Omega\left(\mathrm{3},\mathrm{5}\right)\:>\:\sqrt{\Omega\left(\mathrm{4},\mathrm{5}\right)\centerdot\Omega\left(\mathrm{3},\mathrm{6}\right)} \\ $$
Question Number 162055 Answers: 1 Comments: 0
$$\int{e}^{\mathrm{2x}} \sqrt{\left(\mathrm{1}\:−{e}^{\mathrm{2}{x}} \right)}{dx} \\ $$
Question Number 162054 Answers: 1 Comments: 0
Question Number 162043 Answers: 1 Comments: 2
$$\mathrm{Find}\:\mathrm{valu}\:\mathrm{of}\:\:\boldsymbol{\mathrm{x}}\:\:\mathrm{if}\:\:\mathrm{x}\in\mathbb{R}\: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{9x}\:-\:\mathrm{1}}\:+\:\sqrt{\mathrm{8x}\:-\:\mathrm{1}}\:+\:\sqrt[{\mathrm{4}}]{\mathrm{8x}\:+\:\mathrm{15}}\:-\:\frac{\mathrm{5}}{\mathrm{2}}\:=\:\mathrm{0} \\ $$
Question Number 162042 Answers: 0 Comments: 0
$$\mathrm{let}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\in\mathbb{R}\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \:\geqslant\:\mathrm{a}^{\mathrm{3}} \mathrm{b}\:+\:\mathrm{b}^{\mathrm{3}} \mathrm{c}\:+\:\mathrm{c}^{\mathrm{3}} \mathrm{a} \\ $$
Question Number 162073 Answers: 3 Comments: 0
$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\Omega\:=\int_{−\infty} ^{\:+\infty} \frac{\:{cos}\:\left({x}\right)}{\left(\mathrm{2}+\:\mathrm{2}{x}\:+{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{{e}}\:{cos}\left(\mathrm{1}\right) \\ $$
Question Number 162035 Answers: 2 Comments: 0
$$\left(\underset{\mathrm{1}} {\overset{\mathrm{2014}} {\:}}\right)\:+\:\left(\underset{\mathrm{2}} {\overset{\mathrm{2014}} {\:}}\right)\:+\:\left(\underset{\mathrm{3}} {\overset{\mathrm{2014}} {\:}}\right)\:+\:\ldots+\:\left(\underset{\mathrm{1007}} {\overset{\mathrm{2014}} {\:}}\right)\:=\:? \\ $$
Question Number 162033 Answers: 0 Comments: 2
Question Number 162025 Answers: 0 Comments: 0
$$\:\:\:\: \\ $$$$\:\:\:{write}\:\:{the}\:{taylor}\:{expansion}\:{of}\:: \\ $$$$\:\:\:\:\:\:{f}\left({x}\right)=\:{x}^{\:\mathrm{2}} .\:{cos}\left({x}\right)\:\:\:\:{at}\:\:{x}=\mathrm{1} \\ $$$$\:\:\:\:{then}\:\:\:\:\:\:\:\:{f}^{\:\left(\mathrm{5}\:\right)} \left({x}\right)\:\:{at}\:\:{x}=\mathrm{1}\:\:? \\ $$$$ \\ $$
Question Number 162026 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:{prove}\:{that}.... \\ $$$$\: \\ $$$$\:\:\:\:\:\left(\:\mathrm{1}+\:\frac{\mathrm{1}}{{n}}\:\right)^{\:{n}} \:<\:{e}\:<\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\:\right)^{\:{n}+\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$
Question Number 162016 Answers: 2 Comments: 2
$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\mathrm{3x}\right)}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$
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