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Esta has a directory. it contains n (n<1000) pages where 999991 subscribers are registered and each page contains the same numbers of subscribers. How many pages does this directory has?

$${Esta}\:{has}\:{a} \\ $$$$\:{directory}.\:{it}\:{contains}\: \\ $$$${n}\:\left({n}<\mathrm{1000}\right)\:{pages}\:{where}\: \\ $$$$\mathrm{999991}\:{subscribers}\:{are}\:{registered} \\ $$$${and}\:{each}\:{page}\:{contains}\:{the}\: \\ $$$${same}\:{numbers}\:{of}\:{subscribers}. \\ $$$${How}\:{many}\:{pages}\:{does}\:{this} \\ $$$${directory}\:{has}? \\ $$

Question Number 122641 Answers: 0 Comments: 0

Σ_(n=1) ^∞ (1/(n^4 +1))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{4}} +\mathrm{1}} \\ $$

Question Number 122638 Answers: 0 Comments: 1

Question Number 122518 Answers: 1 Comments: 1

1−(1/5)+(1/7)−(1/(11))+(1/(13))−(1/(17))+...

$$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{7}}−\frac{\mathrm{1}}{\mathrm{11}}+\frac{\mathrm{1}}{\mathrm{13}}−\frac{\mathrm{1}}{\mathrm{17}}+... \\ $$

Question Number 122514 Answers: 2 Comments: 1

Question Number 122336 Answers: 0 Comments: 0

please prove that sup(−A)=−inf(A)

$${please}\:{prove}\:{that}\:{sup}\left(−{A}\right)=−{inf}\left({A}\right) \\ $$

Question Number 122335 Answers: 0 Comments: 0

Question Number 122322 Answers: 4 Comments: 0

∣x+1∣<∣x^2 +2x+2∣ solve for x

$$\mid{x}+\mathrm{1}\mid<\mid{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\mid \\ $$$${solve}\:{for}\:{x} \\ $$

Question Number 122318 Answers: 0 Comments: 0

Question Number 122303 Answers: 1 Comments: 3

((log1)/( (√1)))−((log3)/( (√3)))+((log5)/( (√5)))−((log7)/( (√7)))+..C=((π/4)−(γ/2)−(1/2)log(2π))((1/( (√1)))−(1/( (√3)))+..C) γ =Eulerian constant

$$\frac{{log}\mathrm{1}}{\:\sqrt{\mathrm{1}}}−\frac{{log}\mathrm{3}}{\:\sqrt{\mathrm{3}}}+\frac{{log}\mathrm{5}}{\:\sqrt{\mathrm{5}}}−\frac{{log}\mathrm{7}}{\:\sqrt{\mathrm{7}}}+..{C}=\left(\frac{\pi}{\mathrm{4}}−\frac{\gamma}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}{log}\left(\mathrm{2}\pi\right)\right)\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}+..{C}\right) \\ $$$$\gamma\:=\mathscr{E}{ulerian}\:{constant} \\ $$

Question Number 122302 Answers: 3 Comments: 0

∫_0 ^∞ (dx/((1+x^2 )(4+x^2 )(16+x^2 )(64+x^2 )))

$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{4}+{x}^{\mathrm{2}} \right)\left(\mathrm{16}+{x}^{\mathrm{2}} \right)\left(\mathrm{64}+{x}^{\mathrm{2}} \right)} \\ $$

Question Number 122317 Answers: 0 Comments: 0

1+9((1/4))^4 +17(((1.5)/(4.8)))^4 +25(((1.5.9)/(4.8.12)))^4 +...=(π/(2(√2)(Γ^2 ((3/4)))))

$$\mathrm{1}+\mathrm{9}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} +\mathrm{17}\left(\frac{\mathrm{1}.\mathrm{5}}{\mathrm{4}.\mathrm{8}}\right)^{\mathrm{4}} +\mathrm{25}\left(\frac{\mathrm{1}.\mathrm{5}.\mathrm{9}}{\mathrm{4}.\mathrm{8}.\mathrm{12}}\right)^{\mathrm{4}} +...=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}}\left(\Gamma^{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right)} \\ $$

Question Number 122176 Answers: 2 Comments: 2

∫_0 ^(π/2) ((tanx))^(1/7) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt[{\mathrm{7}}]{{tanx}}\:{dx} \\ $$

Question Number 122377 Answers: 1 Comments: 4

Question Number 122069 Answers: 1 Comments: 0

∫_0 ^1 (1/( ((x^(96) −x^(97) ))^(1/(97)) ))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt[{\mathrm{97}}]{{x}^{\mathrm{96}} −{x}^{\mathrm{97}} }}{dx} \\ $$

Question Number 122023 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (((−1)^n F_n )/7^n ) (F_n denotes Fibbonocci sequence)

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {F}_{{n}} }{\mathrm{7}^{{n}} }\:\:\:\:\:\:\left({F}_{{n}} \:{denotes}\:{Fibbonocci}\:{sequence}\right) \\ $$

Question Number 122013 Answers: 0 Comments: 6

∫_0 ^(π/(24)) log(tanθ)dθ Problem source: brilliant

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{24}}} {log}\left({tan}\theta\right){d}\theta \\ $$$${Problem}\:{source}:\:{brilliant} \\ $$

Question Number 121983 Answers: 1 Comments: 2

∫_0 ^(π/2) sin^n x dx (In closed form) (n∈N)

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{{n}} {x}\:{dx}\:\left({In}\:{closed}\:{form}\right)\:\:\left({n}\in\mathbb{N}\right) \\ $$

Question Number 121914 Answers: 0 Comments: 1

∫_0 ^(π/2) (sinθ)^(−(1/n)) dθ

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sin}\theta\right)^{−\frac{\mathrm{1}}{{n}}} {d}\theta \\ $$

Question Number 121739 Answers: 1 Comments: 0

(1^(13) /(e^(2π) −1))+(2^(13) /(e^(4π) −1))+(3^(13) /(e^(6π) −1))+....

$$\frac{\mathrm{1}^{\mathrm{13}} }{{e}^{\mathrm{2}\pi} −\mathrm{1}}+\frac{\mathrm{2}^{\mathrm{13}} }{{e}^{\mathrm{4}\pi} −\mathrm{1}}+\frac{\mathrm{3}^{\mathrm{13}} }{{e}^{\mathrm{6}\pi} −\mathrm{1}}+.... \\ $$

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