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Question Number 154553    Answers: 0   Comments: 0

Π_(n=1) ^∞ (( Γ(n+ (1/n^2 )) )/( Γ(n+ (1/n)) ))

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{\:\Gamma\left({n}+\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\:}{\:\Gamma\left({n}+\:\frac{\mathrm{1}}{{n}}\right)\:} \\ $$$$\: \\ $$

Question Number 154552    Answers: 2   Comments: 28

Question Number 154549    Answers: 4   Comments: 1

A particle is projected with velocity 2(√(gh)) so that it just clears two walls of equal heigh(h) in the t_1 and t_(2 ) respectively.The two walls are at a distance of 2h from each other.If time passing between the two walls is 2(√(h/g)) show that (i) angle projected 60^° (ii)t_1 +t_2 =2(√((3h)/g))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\:\mathrm{with}\:\mathrm{velocity} \\ $$$$\mathrm{2}\sqrt{\mathrm{gh}}\:\:\:\mathrm{so}\:\mathrm{that}\:\:\mathrm{it}\:\mathrm{just}\:\mathrm{clears}\:\mathrm{two} \\ $$$$\mathrm{walls}\:\mathrm{of}\:\mathrm{equal}\:\mathrm{heigh}\left(\mathrm{h}\right)\:\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{t}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{t}_{\mathrm{2}\:} \:\mathrm{respectively}.\mathrm{The}\:\mathrm{two} \\ $$$$\mathrm{walls}\:\mathrm{are}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\:\mathrm{2h}\:\:\mathrm{from} \\ $$$$\mathrm{each}\:\mathrm{other}.\mathrm{If}\:\mathrm{time}\:\mathrm{passing}\: \\ $$$$\mathrm{between}\:\mathrm{the}\:\mathrm{two}\:\mathrm{walls}\:\mathrm{is}\:\mathrm{2}\sqrt{\frac{\mathrm{h}}{\mathrm{g}}} \\ $$$$\mathrm{show}\:\mathrm{that}\:\left(\mathrm{i}\right)\:\mathrm{angle}\:\mathrm{projected}\:\mathrm{60}^{°} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{ii}\right)\mathrm{t}_{\mathrm{1}} +\mathrm{t}_{\mathrm{2}} =\mathrm{2}\sqrt{\frac{\mathrm{3h}}{\mathrm{g}}} \\ $$$$ \\ $$

Question Number 154547    Answers: 1   Comments: 0

A particle is projected inside the tunnel which is 4m high.if the initial speed is V_o .show that the maximum range inside the tunnel is given by R=4(√2) (√((V_o ^2 /g)−8))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{tunnel} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{4m}\:\mathrm{high}.\mathrm{if}\:\:\mathrm{the}\:\mathrm{initial}\:\mathrm{speed} \\ $$$$\mathrm{is}\:\mathrm{V}_{\mathrm{o}} .\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{range}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{tunnel}\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{by}\:\:\:\:\:\:\mathrm{R}=\mathrm{4}\sqrt{\mathrm{2}}\:\sqrt{\frac{\mathrm{V}_{\mathrm{o}} ^{\mathrm{2}} }{\mathrm{g}}−\mathrm{8}} \\ $$

Question Number 154573    Answers: 2   Comments: 1

Question Number 154544    Answers: 0   Comments: 0

determinant (((prove that)),((Σ_(n=1) ^∞ ((H_n H_n ^((2)) )/n^3 )+Σ_(n=1) ^∞ ((H_n H_n ^((3)) )/n^2 )=((21)/8)𝛇(6)+𝛇^2 (3)))) by Math.Amin 11.fb.96

$$\begin{array}{|c|c|}{\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}}\\{\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} ^{\left(\mathrm{2}\right)} }{\boldsymbol{\mathrm{n}}^{\mathrm{3}} }+\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} ^{\left(\mathrm{3}\right)} }{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }=\frac{\mathrm{21}}{\mathrm{8}}\boldsymbol{\zeta}\left(\mathrm{6}\right)+\boldsymbol{\zeta}^{\mathrm{2}} \left(\mathrm{3}\right)}\\\hline\end{array} \\ $$$${by}\:{Math}.{Amin}\:\:\mathrm{11}.{fb}.\mathrm{96} \\ $$

Question Number 154542    Answers: 1   Comments: 0

Question Number 154530    Answers: 0   Comments: 0

Question Number 154516    Answers: 0   Comments: 0

Question Number 154514    Answers: 0   Comments: 1

Question Number 154586    Answers: 1   Comments: 0

Question Number 154507    Answers: 2   Comments: 1

Question Number 154649    Answers: 0   Comments: 2

if n ∈ N^(>2) prove that [((n)^(1/3) + ((n + 2))^(1/3) )^3 ] + 1 = 0 (mod 8)

$$\mathrm{if}\:\:\mathrm{n}\:\in\:\mathbb{N}^{>\mathrm{2}} \:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\left[\left(\sqrt[{\mathrm{3}}]{\mathrm{n}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{n}\:+\:\mathrm{2}}\:\right)^{\mathrm{3}} \right]\:+\:\mathrm{1}\:=\:\mathrm{0}\:\left(\mathrm{mod}\:\mathrm{8}\right) \\ $$

Question Number 154620    Answers: 0   Comments: 0

Question Number 154497    Answers: 2   Comments: 0

Solve the equations: a) 2 (√(2x^3 - x)) = 3x^2 - 3x + 2 b) (√((x^4 + 16)/2)) + (√(2(x^2 + 4))) = 3x + 2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equations}: \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\:\:\:\mathrm{2}\:\sqrt{\mathrm{2x}^{\mathrm{3}} \:-\:\mathrm{x}}\:=\:\mathrm{3x}^{\mathrm{2}} \:-\:\mathrm{3x}\:+\:\mathrm{2} \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\:\:\:\sqrt{\frac{\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{16}}{\mathrm{2}}}\:+\:\sqrt{\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4}\right)}\:=\:\mathrm{3x}\:+\:\mathrm{2} \\ $$

Question Number 154495    Answers: 1   Comments: 0

if S_n (t) = n^(1-t) ((((n+1)^(2t) )/(((((n+1)!))^(1/(n+1)) )^t )) - (n^(2t) /((((n!))^(1/n) )^t ))) with t>0 then lim_(n→∞) S_n (t) = te^t

$$\mathrm{if}\:\:\mathrm{S}_{\boldsymbol{\mathrm{n}}} \left(\mathrm{t}\right)\:=\:\mathrm{n}^{\mathrm{1}-\boldsymbol{\mathrm{t}}} \:\left(\frac{\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{2}\boldsymbol{\mathrm{t}}} }{\left(\sqrt[{\boldsymbol{\mathrm{n}}+\mathrm{1}}]{\left(\mathrm{n}+\mathrm{1}\right)!}\right)^{\boldsymbol{\mathrm{t}}} }\:-\:\frac{\mathrm{n}^{\mathrm{2}\boldsymbol{\mathrm{t}}} }{\left(\sqrt[{\boldsymbol{\mathrm{n}}}]{\mathrm{n}!}\right)^{\boldsymbol{\mathrm{t}}} }\right) \\ $$$$\mathrm{with}\:\:\mathrm{t}>\mathrm{0} \\ $$$$\mathrm{then}\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}S}_{\boldsymbol{\mathrm{n}}} \left(\mathrm{t}\right)\:=\:\mathrm{te}^{\boldsymbol{\mathrm{t}}} \\ $$

Question Number 154493    Answers: 0   Comments: 0

If a;b;c>0 and n∈N^+ then: ((a^(2n) + b^(2n) + c^(2n) )/(a^n b^n + b^n c^n + c^n a^n )) ≥ ((√(3∙(a^2 + b^2 + c^2 )))/(a + b + c))

$$\mathrm{If}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{n}\in\mathbb{N}^{+} \:\:\mathrm{then}: \\ $$$$\frac{\mathrm{a}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{c}^{\mathrm{2}\boldsymbol{\mathrm{n}}} }{\mathrm{a}^{\boldsymbol{\mathrm{n}}} \mathrm{b}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}} \mathrm{c}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{c}^{\boldsymbol{\mathrm{n}}} \mathrm{a}^{\boldsymbol{\mathrm{n}}} }\:\geqslant\:\frac{\sqrt{\mathrm{3}\centerdot\left(\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \right)}}{\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}} \\ $$

Question Number 154478    Answers: 1   Comments: 0

prove that # ∫_0 ^( ∞) (( sin^( 3) ( x ).ln( x ))/x) dx =^? (π/8) (−2γ +ln(3)) .....■ m.n

$$ \\ $$$$ \\ $$$$\:\:{prove}\:{that}\:# \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{3}} \left(\:{x}\:\right).{ln}\left(\:{x}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\pi}{\mathrm{8}}\:\left(−\mathrm{2}\gamma\:+{ln}\left(\mathrm{3}\right)\right)\:.....\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\: \\ $$$$ \\ $$

Question Number 154476    Answers: 3   Comments: 0

soit:y′′−3y′−4y=3e^(3x ) avec , f(o)=−(1/2) et f′(0)=4 alors f(1)=?

$${soit}:{y}''−\mathrm{3}{y}'−\mathrm{4}{y}=\mathrm{3}{e}^{\mathrm{3}{x}\:} \:{avec}\:, \\ $$$${f}\left({o}\right)=−\frac{\mathrm{1}}{\mathrm{2}}\:{et}\:{f}'\left(\mathrm{0}\right)=\mathrm{4} \\ $$$${alors}\:{f}\left(\mathrm{1}\right)=? \\ $$$$ \\ $$

Question Number 154475    Answers: 2   Comments: 4

Question Number 154467    Answers: 1   Comments: 0

Π_(n=1) ^∞ (((1+ (1/n))^(1/2) )/(1+ (1/(2n))))

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{\left(\mathrm{1}+\:\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} }{\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2}{n}}} \\ $$$$\: \\ $$

Question Number 154465    Answers: 0   Comments: 2

Question Number 154458    Answers: 1   Comments: 0

Question Number 154456    Answers: 1   Comments: 2

S=90^2 +91^2 +.....+100^2 = ??

$${S}=\mathrm{90}^{\mathrm{2}} +\mathrm{91}^{\mathrm{2}} +.....+\mathrm{100}^{\mathrm{2}} \:\:=\:\:?? \\ $$

Question Number 154469    Answers: 1   Comments: 0

prove:∫_0 ^( 1) ln (4− 2x +x^( 2) )dx =2ln((2/e)) +(π/( (√3)))

$$ \\ $$$${prove}:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\:\left(\mathrm{4}−\:\mathrm{2}{x}\:+{x}^{\:\mathrm{2}} \right){dx}\:=\mathrm{2}{ln}\left(\frac{\mathrm{2}}{{e}}\right)\:+\frac{\pi}{\:\sqrt{\mathrm{3}}} \\ $$$$ \\ $$

Question Number 154444    Answers: 2   Comments: 1

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