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Question Number 116166    Answers: 1   Comments: 0

∀n∈N^∗ , suppose u_n =(5sin(1/n^2 )+(1/5)cos n)^n Prove that lim_(n→+∞) u_n =0

$$\forall{n}\in\mathbb{N}^{\ast} ,\:\mathrm{suppose}\:{u}_{{n}} =\left(\mathrm{5sin}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}}\mathrm{cos}\:{n}\right)^{{n}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{n}\rightarrow+\infty} {\mathrm{lim}}{u}_{{n}} =\mathrm{0} \\ $$

Question Number 116162    Answers: 1   Comments: 0

Σ_(n=0) ^∞ ((2n+1)/(16^n (n^2 +3n+2))) (((2n)),(n) )^2 =(8/(3π)) m.n.july 1970.

$$\:\: \\ $$$$ \\ $$$$\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{16}^{{n}} \left({n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}\right)}\:\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}^{\mathrm{2}} \:=\frac{\mathrm{8}}{\mathrm{3}\pi}\: \\ $$$$ \\ $$$${m}.{n}.{july}\:\mathrm{1970}. \\ $$$$\: \\ $$

Question Number 116163    Answers: 1   Comments: 0

I need a general rule for factorising 1) a^n +b^n 2) a^n −b^n when n is even and when n is odd.

$$\mathrm{I}\:\mathrm{need}\:\mathrm{a}\:\mathrm{general}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{factorising} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{a}^{\mathrm{n}} +\mathrm{b}^{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} \\ $$$$\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}. \\ $$

Question Number 116196    Answers: 6   Comments: 0

please solve : ∫_0 ^( (π/4)) tan^9 (x)dx =???

$$\:\:\:\:\:{please}\:{solve}\:: \\ $$$$\:\:\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {tan}^{\mathrm{9}} \left({x}\right){dx}\:=??? \\ $$

Question Number 116149    Answers: 0   Comments: 1

Question Number 116138    Answers: 3   Comments: 1

If α and β are the roots of the quadratic equation x^2 −10x+2=0 and α >β, find: (i) (1/β)−(1/α) (ii)α^3 −β^3

$$\mathrm{If}\:\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{10x}+\mathrm{2}=\mathrm{0}\:\mathrm{and}\:\alpha\:>\beta,\:\mathrm{find}: \\ $$$$\left(\mathrm{i}\right)\:\frac{\mathrm{1}}{\beta}−\frac{\mathrm{1}}{\alpha} \\ $$$$\left(\mathrm{ii}\right)\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} \\ $$

Question Number 116136    Answers: 3   Comments: 0

y′ =(y/x) +((2x^3 cos (x^2 ))/y) where y((√π)) = 0

$$\mathrm{y}'\:=\frac{\mathrm{y}}{\mathrm{x}}\:+\frac{\mathrm{2x}^{\mathrm{3}} \:\mathrm{cos}\:\left(\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{y}} \\ $$$$\mathrm{where}\:\mathrm{y}\left(\sqrt{\pi}\right)\:=\:\mathrm{0} \\ $$

Question Number 116114    Answers: 1   Comments: 0

Question Number 116113    Answers: 1   Comments: 0

what′s the exponent of 12 in the expansion of 100!

$$\boldsymbol{\mathrm{what}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{exponent}}\:\boldsymbol{\mathrm{of}}\:\mathrm{12}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}}\:\mathrm{100}! \\ $$

Question Number 116112    Answers: 2   Comments: 0

show that ∫_( 0) ^( ∞) ((lnx)/(1+x^2 ))dx = 0

$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\int_{\:\mathrm{0}} ^{\:\infty} \frac{\boldsymbol{\mathrm{lnx}}}{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}\:=\:\mathrm{0} \\ $$$$ \\ $$

Question Number 116226    Answers: 2   Comments: 1

(1/(2+(√2))) +(1/(3(√2)+2(√3))) +(1/(4(√3)+3(√4)))+...+(1/(100(√(99))+99(√(100))))

$$\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}}}\:+\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{4}}}+...+\frac{\mathrm{1}}{\mathrm{100}\sqrt{\mathrm{99}}+\mathrm{99}\sqrt{\mathrm{100}}} \\ $$

Question Number 116123    Answers: 0   Comments: 0

Study according to the values of the real α the convergence of the integral ∫_α ^(+∞) ((ln∣x∣)/( ((x(x+1)))^(1/3) ))dx

$$ \\ $$$$\mathrm{Study}\:\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{real}\:\alpha\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{integral}\:\:\int_{\alpha} ^{+\infty} \frac{{ln}\mid{x}\mid}{\:\sqrt[{\mathrm{3}}]{{x}\left({x}+\mathrm{1}\right)}}{dx} \\ $$

Question Number 116105    Answers: 2   Comments: 0

solve the Cauchy-Euler Differential Equation by substituting x=e^t x^3 (d^3 y/dx^3 ) + 2x^2 (d^2 y/dx^2 ) + 2y = 10x + ((10)/x)

$${solve}\:{the}\:{Cauchy}-{Euler}\: \\ $$$${Differential}\:{Equation}\:{by} \\ $$$${substituting}\:{x}={e}^{{t}} \\ $$$${x}^{\mathrm{3}} \:\frac{{d}^{\mathrm{3}} {y}}{{dx}^{\mathrm{3}} }\:+\:\mathrm{2}{x}^{\mathrm{2}} \:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{2}{y}\:=\:\mathrm{10}{x}\:+\:\frac{\mathrm{10}}{{x}} \\ $$$$ \\ $$

Question Number 116098    Answers: 0   Comments: 0

1)calculate f(x)=∫_0 ^(2π) (dθ/(x^2 −2x cosθ +1)) 0<θ<(π/2) 2)explicite ∫_0 ^(2π) ((cosθ)/((x^2 −2xcosθ +1)^2 ))dθ

$$\left.\mathrm{1}\right)\mathrm{calculate}\:\mathrm{f}\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{\mathrm{d}\theta}{\mathrm{x}^{\mathrm{2}} −\mathrm{2x}\:\mathrm{cos}\theta\:+\mathrm{1}}\:\:\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\mathrm{explicite}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{\mathrm{cos}\theta}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{d}\theta \\ $$

Question Number 116097    Answers: 3   Comments: 0

calculate ∫_0 ^∞ ((lnx)/(x^4 +1))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{lnx}}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{1}}\mathrm{dx}\: \\ $$

Question Number 116096    Answers: 1   Comments: 0

find ∫_0 ^∞ ((lnx)/(x^2 −i))dx (i=(√(−1)))

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{lnx}}{\mathrm{x}^{\mathrm{2}} −\mathrm{i}}\mathrm{dx}\:\:\:\:\:\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 116094    Answers: 1   Comments: 0

Question Number 116093    Answers: 2   Comments: 0

Find the equation of a circle which touches x−axis and the line y=x in the 1^(st) quadrant. Determine its centre and radius if it touches the line y+x=4.

$${Find}\:{the}\:{equation}\:{of}\:{a}\:{circle}\:{which}\:{touches} \\ $$$${x}−{axis}\:{and}\:{the}\:{line}\:{y}={x}\:{in}\:{the}\:\mathrm{1}^{{st}} \:{quadrant}. \\ $$$${Determine}\:{its}\:{centre}\:{and}\:{radius}\:{if}\:{it}\:{touches}\:{the}\:{line}\:{y}+{x}=\mathrm{4}. \\ $$

Question Number 116092    Answers: 1   Comments: 2

Solve for x and y if x^y =36

$${Solve}\:{for}\:{x}\:{and}\:{y}\:{if} \\ $$$${x}^{{y}} =\mathrm{36} \\ $$

Question Number 116091    Answers: 2   Comments: 0

Is there a formular to tell how many times a digit occur in an interval. e.g. How many times digits 2 occur between 1 − 100

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{formular}\:\mathrm{to}\:\mathrm{tell}\:\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{a}\:\mathrm{digit}\:\mathrm{occur}\:\mathrm{in}\:\mathrm{an}\:\mathrm{interval}. \\ $$$$ \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{digits}\:\:\mathrm{2}\:\:\mathrm{occur}\:\mathrm{between}\:\:\mathrm{1}\:−\:\mathrm{100} \\ $$

Question Number 116090    Answers: 1   Comments: 0

what is 3(7/8)hrs?

$${what}\:{is}\:\mathrm{3}\frac{\mathrm{7}}{\mathrm{8}}{hrs}? \\ $$

Question Number 116087    Answers: 0   Comments: 2

soit f la fonction de^ finie sur[0,2] par f(x)=3 si x∈[0,2]∩Q f(x)=1 si x∈[0,2]∩R\Q

$${soit}\:{f}\:{la}\:{fonction}\:{d}\acute {{e}finie}\:{sur}\left[\mathrm{0},\mathrm{2}\right]\:{par}\: \\ $$$${f}\left({x}\right)=\mathrm{3}\:\:\:{si}\:{x}\in\left[\mathrm{0},\mathrm{2}\right]\cap\mathbb{Q} \\ $$$${f}\left({x}\right)=\mathrm{1}\:\:\:{si}\:{x}\in\left[\mathrm{0},\mathrm{2}\right]\cap\mathbb{R}\backslash\mathbb{Q} \\ $$

Question Number 116083    Answers: 0   Comments: 0

Given X ∣ Θ=θ ∽ Uniform(0,θ) and Θ ∽ Uniform(20,40). Find the cdf of X, F_X (x) for x∈[0,20) and x∈[20,40)

$$\mathrm{Given}\:{X}\:\mid\:\Theta=\theta\:\backsim\:{Uniform}\left(\mathrm{0},\theta\right)\:\mathrm{and}\: \\ $$$$\Theta\:\backsim\:{Uniform}\left(\mathrm{20},\mathrm{40}\right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cdf}\:\mathrm{of}\:{X},\:\:{F}_{{X}} \left({x}\right)\:\mathrm{for}\:{x}\in\left[\mathrm{0},\mathrm{20}\right)\:\mathrm{and}\:{x}\in\left[\mathrm{20},\mathrm{40}\right) \\ $$

Question Number 116085    Answers: 0   Comments: 1

6+log_(3/2) {(1/(3(√2)))(√(4−(1/(3(√2)))(√(4−(1/(3(√2)))(√(4−(1/(3(√2)))∙∙∙))))))}= ?

$$\mathrm{6}+\mathrm{log}_{\frac{\mathrm{3}}{\mathrm{2}}} \left\{\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\sqrt{\mathrm{4}−\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\sqrt{\mathrm{4}−\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\sqrt{\mathrm{4}−\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}}\centerdot\centerdot\centerdot}}}\right\}=\:? \\ $$

Question Number 116079    Answers: 1   Comments: 0

prove that Re=((ρ∙v∙d)/μ) renulds number

$${prove}\:{that}\:{Re}=\frac{\rho\centerdot{v}\centerdot{d}}{\mu}\:\:\:\:\:{renulds}\:{number} \\ $$

Question Number 116078    Answers: 1   Comments: 0

prove that Fr=(v^2 /(gh)) froude numer

$${prove}\:{that}\:\:\:{Fr}=\frac{{v}^{\mathrm{2}} }{{gh}}\:\:\:\:{froude}\:{numer} \\ $$

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