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Question Number 123947 Answers: 0 Comments: 0

((9^2 −1^2 )/(9^2 −2^2 )).((18^2 −1^2 )/(18^2 −0^2 )).((27^2 −1^2 )/(27^2 −2^2 )).((36^2 −1^2 )/(36^2 −0^2 )).....

$$\frac{\mathrm{9}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }{\mathrm{9}^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} }.\frac{\mathrm{18}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }{\mathrm{18}^{\mathrm{2}} −\mathrm{0}^{\mathrm{2}} }.\frac{\mathrm{27}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }{\mathrm{27}^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} }.\frac{\mathrm{36}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }{\mathrm{36}^{\mathrm{2}} −\mathrm{0}^{\mathrm{2}} }..... \\ $$

Question Number 123845 Answers: 1 Comments: 1

Question Number 123823 Answers: 1 Comments: 0

Question Number 123807 Answers: 0 Comments: 1

((((1/2))!(1)!)/(((5/2))!))−((((3/2))!(2)!)/(((9/2))!))+((((5/2))!3!)/((((13)/2))!))−((((7/2))!4!)/((((17)/2))!))+....

$$\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)!\left(\mathrm{1}\right)!}{\left(\frac{\mathrm{5}}{\mathrm{2}}\right)!}−\frac{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)!\left(\mathrm{2}\right)!}{\left(\frac{\mathrm{9}}{\mathrm{2}}\right)!}+\frac{\left(\frac{\mathrm{5}}{\mathrm{2}}\right)!\mathrm{3}!}{\left(\frac{\mathrm{13}}{\mathrm{2}}\right)!}−\frac{\left(\frac{\mathrm{7}}{\mathrm{2}}\right)!\mathrm{4}!}{\left(\frac{\mathrm{17}}{\mathrm{2}}\right)!}+.... \\ $$

Question Number 123776 Answers: 0 Comments: 0

Another unanswered question ((log1)/( (√1)))−((log3)/( (√3)))+((log5)/( (√5)))−((log7)/( (√7)))+..=((π/4)−(γ/2)−(1/2)log(2π))((1/( (√1)))−(1/( (√3)))+(1/( (√5)))−.)

$${Another}\:{unanswered}\:{question} \\ $$$$\frac{{log}\mathrm{1}}{\:\sqrt{\mathrm{1}}}−\frac{{log}\mathrm{3}}{\:\sqrt{\mathrm{3}}}+\frac{{log}\mathrm{5}}{\:\sqrt{\mathrm{5}}}−\frac{{log}\mathrm{7}}{\:\sqrt{\mathrm{7}}}+..=\left(\frac{\pi}{\mathrm{4}}−\frac{\gamma}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}{log}\left(\mathrm{2}\pi\right)\right)\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}−.\right) \\ $$

Question Number 123794 Answers: 0 Comments: 0

...mathematical analysis... suppose (X_1 ,τ_1 ) and (X_2 ,τ_2 ) are two topological spaces. prove f:(X_1 ,τ_1 )→(X_2 ,τ_2 ) is a continuous function if only if for any subset A⊆X_1 : f(cl(A))⊆cl(f(A))

$$\:\:\:\:\:\:\:\:\:\:\:...{mathematical}\:\:{analysis}... \\ $$$$\:{suppose}\:\:\left({X}_{\mathrm{1}} ,\tau_{\mathrm{1}} \right)\:{and}\:\left({X}_{\mathrm{2}} ,\tau_{\mathrm{2}} \right) \\ $$$$\:{are}\:{two}\:{topological}\:{spaces}. \\ $$$${prove}\:\:{f}:\left({X}_{\mathrm{1}} ,\tau_{\mathrm{1}} \right)\rightarrow\left({X}_{\mathrm{2}} ,\tau_{\mathrm{2}} \right)\:{is}\: \\ $$$${a}\:{continuous}\:{function}\:{if}\:{only} \\ $$$${if}\:\:{for}\:{any}\:{subset}\:{A}\subseteq{X}_{\mathrm{1}} \:: \\ $$$$\:\:\:\:\:\:\:{f}\left({cl}\left({A}\right)\right)\subseteq{cl}\left({f}\left({A}\right)\right) \\ $$$$\: \\ $$

Question Number 123694 Answers: 0 Comments: 1

sinh((1/1^2 ))+sinh((1/2^2 ))+sinh((1/3^2 ))+....

$${sinh}\left(\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }\right)+{sinh}\left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)+{sinh}\left(\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\right)+.... \\ $$

Question Number 123653 Answers: 1 Comments: 0

Question Number 123651 Answers: 1 Comments: 0

Question Number 123650 Answers: 0 Comments: 4

1−(1/4)+(1/8)−(1/(10))+(1/(14))−(1/(16))+...

$$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}−\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{14}}−\frac{\mathrm{1}}{\mathrm{16}}+... \\ $$

Question Number 123620 Answers: 0 Comments: 0

((coth(π))/1^7 )+((coth(2π))/2^7 )+((coth(3π))/3^7 )+...

$$\frac{{coth}\left(\pi\right)}{\mathrm{1}^{\mathrm{7}} }+\frac{{coth}\left(\mathrm{2}\pi\right)}{\mathrm{2}^{\mathrm{7}} }+\frac{{coth}\left(\mathrm{3}\pi\right)}{\mathrm{3}^{\mathrm{7}} }+... \\ $$

Question Number 123460 Answers: 0 Comments: 1

sinh^(−1) (1)+sinh^(−1) ((1/2^2 ))+sinh^(−1) ((1/3^2 ))+sinh^(−1) ((1/4^2 ))+....

$${sinh}^{−\mathrm{1}} \left(\mathrm{1}\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{2}} }\right)+.... \\ $$$$ \\ $$

Question Number 123370 Answers: 0 Comments: 2

((√1)/1)+((√2)/5)+((√3)/(10))+((√4)/(17))+((√5)/(37))+...

$$\frac{\sqrt{\mathrm{1}}}{\mathrm{1}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{5}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{10}}+\frac{\sqrt{\mathrm{4}}}{\mathrm{17}}+\frac{\sqrt{\mathrm{5}}}{\mathrm{37}}+... \\ $$

Question Number 123336 Answers: 1 Comments: 0

A mercury based barometer reads 760mmhg at the base of a mountain and 704.0mmhg at the top. Find the height of the mountain if the density of mercury and air are respectively 13600kgm^(−3) and 1.25kgm^(−3) (g=10ms^(−2) )

$$ \\ $$$$\mathrm{A}\:\mathrm{mercury}\:\mathrm{based}\:\mathrm{barometer}\:\mathrm{reads} \\ $$$$\mathrm{760mmhg}\:\mathrm{at}\:\mathrm{the}\:\mathrm{base}\:\mathrm{of}\:\mathrm{a}\:\mathrm{mountain}\:\mathrm{and} \\ $$$$\mathrm{704}.\mathrm{0mmhg}\:\mathrm{at}\:\mathrm{the}\:\mathrm{top}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{height}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{mountain}\:\mathrm{if}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\: \\ $$$$\mathrm{mercury}\:\mathrm{and}\:\mathrm{air}\:\mathrm{are}\:\mathrm{respectively} \\ $$$$\mathrm{13600kgm}^{−\mathrm{3}} \:\:\mathrm{and}\:\mathrm{1}.\mathrm{25kgm}^{−\mathrm{3}} \:\left(\mathrm{g}=\mathrm{10ms}^{−\mathrm{2}} \right) \\ $$

Question Number 123326 Answers: 1 Comments: 1

Σ_(n=1) ^∞ (1/(n^3 +n^2 +n+1))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{3}} +{n}^{\mathrm{2}} +{n}+\mathrm{1}} \\ $$

Question Number 123248 Answers: 2 Comments: 3

Find the value of sin((π/9)) in exact form

$${Find}\:{the}\:{value}\:{of}\:{sin}\left(\frac{\pi}{\mathrm{9}}\right)\:\:{in}\:{exact}\:{form} \\ $$

Question Number 123179 Answers: 0 Comments: 3

(1^7 /(e^(2π) −1))+(2^7 /(e^(4π) −1))+(3^7 /(e^(6π) −1))+....

$$\frac{\mathrm{1}^{\mathrm{7}} }{{e}^{\mathrm{2}\pi} −\mathrm{1}}+\frac{\mathrm{2}^{\mathrm{7}} }{{e}^{\mathrm{4}\pi} −\mathrm{1}}+\frac{\mathrm{3}^{\mathrm{7}} }{{e}^{\mathrm{6}\pi} −\mathrm{1}}+.... \\ $$

Question Number 123167 Answers: 1 Comments: 1

Σ_(n=1) ^∞ ((√n)/(n^2 +n+1))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\sqrt{{n}}}{{n}^{\mathrm{2}} +{n}+\mathrm{1}} \\ $$

Question Number 123135 Answers: 0 Comments: 0

∫_0 ^∞ ((ta−bt^3 )/((1+ct^2 )(e^(2πt) −1)))dt

$$\int_{\mathrm{0}} ^{\infty} \frac{{ta}−{bt}^{\mathrm{3}} }{\left(\mathrm{1}+{ct}^{\mathrm{2}} \right)\left({e}^{\mathrm{2}\pi{t}} −\mathrm{1}\right)}{dt} \\ $$

Question Number 122996 Answers: 0 Comments: 1

Σ_(n=1) ^∞ (n/(n^4 +1))

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

Question Number 122960 Answers: 0 Comments: 2

(1/(1+(1/(2+(3^2 /(2+(5^2 /(2+(7^2 /(....))))))))))=(π/4)

$$\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{2}+\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{2}+\frac{\mathrm{7}^{\mathrm{2}} }{....}}}}}=\frac{\pi}{\mathrm{4}}\:\: \\ $$$$ \\ $$

Question Number 122917 Answers: 0 Comments: 0

Σ_(n=1) ^∞ ((tan^(−1) n)/(n^2 +1))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{tan}^{−\mathrm{1}} {n}}{{n}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 122862 Answers: 1 Comments: 0

∫_0 ^∞ (x^7 /((1+x)^(10) ))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{7}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{10}} }{dx} \\ $$

Question Number 122740 Answers: 0 Comments: 3

Prove that tanx=(2/(π−2x))−(2/(π+2x))+(2/(3x−2π))−(2/(3x+2π))+(2/(5x−2π))−(2/(5x+2π))+....

$${Prove}\:{that} \\ $$$${tanx}=\frac{\mathrm{2}}{\pi−\mathrm{2}{x}}−\frac{\mathrm{2}}{\pi+\mathrm{2}{x}}+\frac{\mathrm{2}}{\mathrm{3}{x}−\mathrm{2}\pi}−\frac{\mathrm{2}}{\mathrm{3}{x}+\mathrm{2}\pi}+\frac{\mathrm{2}}{\mathrm{5}{x}−\mathrm{2}\pi}−\frac{\mathrm{2}}{\mathrm{5}{x}+\mathrm{2}\pi}+.... \\ $$

Question Number 122668 Answers: 0 Comments: 0

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))

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

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