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Question Number 34717    Answers: 0   Comments: 1

let I_n = ∫∫_([(1/n),n]^2 ) (((√(xy)) dxdy)/(2 +x^2 +y^2 )) find lim I_n when n→+∞.

$${let}\:{I}_{{n}} =\:\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\right]^{\mathrm{2}} } \:\:\:\:\:\frac{\sqrt{{xy}}\:{dxdy}}{\mathrm{2}\:+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} } \\ $$$${find}\:{lim}\:{I}_{{n}} \:{when}\:{n}\rightarrow+\infty. \\ $$

Question Number 34716    Answers: 0   Comments: 1

calculate ∫∫_w x(√(x^2 +y^2 )) dxdy w ={(x,y)/ x^2 +y^2 ≤3 }

$${calculate}\:\int\int_{{w}} {x}\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\:{dxdy} \\ $$$${w}\:=\left\{\left({x},{y}\right)/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{3}\:\right\}\: \\ $$

Question Number 34715    Answers: 0   Comments: 0

calculate ∫∫_(0≤x≤y≤1) ((dxdy)/((x^2 +1)(y^2 +3))) .

$${calculate}\:\int\int_{\mathrm{0}\leqslant{x}\leqslant{y}\leqslant\mathrm{1}} \:\:\:\frac{{dxdy}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({y}^{\mathrm{2}} \:+\mathrm{3}\right)}\:. \\ $$

Question Number 34714    Answers: 0   Comments: 1

calculate ∫∫_(x^2 +2y^2 ≤1) (x^2 −y^2 )dxdy

$${calculate}\:\int\int_{{x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} \:\leqslant\mathrm{1}} \left({x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} \right){dxdy} \\ $$

Question Number 34713    Answers: 0   Comments: 1

let a>0 calculate ∫∫_(x^2 +y^2 ≤3) (1/(2 +x^2 +y^2 ))dxdy.

$${let}\:{a}>\mathrm{0}\:\:{calculate}\:\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{3}} \:\frac{\mathrm{1}}{\mathrm{2}\:+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy}. \\ $$

Question Number 34707    Answers: 0   Comments: 2

given the sum of the first n terms of an A.P is n^2 the sum of of the first 2n terms of the same A.P is n^2 +n. show that the sum of the first 4n terms is 4n^2 −8n+4.

$$\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{A}}.\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\: \\ $$$$\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{A}}.\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}. \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}} \\ $$$$\mathrm{4}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{is}}\:\mathrm{4}\boldsymbol{\mathrm{n}}^{\mathrm{2}} −\mathrm{8}\boldsymbol{\mathrm{n}}+\mathrm{4}. \\ $$

Question Number 34703    Answers: 1   Comments: 0

A man pushes a box of 40kg up an incline of 15°,if the man applies a horizontal force 200N and the box moves up the plane a distance of 20m at a constant velocity and the coefficient of friction is 0.1, find a)the workdone by the man on the box b)workdone against friction

$${A}\:{man}\:{pushes}\:{a}\:{box}\:{of}\:\mathrm{40}{kg}\:{up}\:{an} \\ $$$${incline}\:{of}\:\mathrm{15}°,{if}\:{the}\:{man}\:{applies} \\ $$$${a}\:{horizontal}\:{force}\:\mathrm{200}{N}\:{and}\:{the} \\ $$$${box}\:{moves}\:{up}\:{the}\:{plane}\:{a}\:{distance} \\ $$$${of}\:\mathrm{20}{m}\:{at}\:{a}\:{constant}\:{velocity}\:{and} \\ $$$${the}\:{coefficient}\:{of}\:{friction}\:{is}\:\mathrm{0}.\mathrm{1}, \\ $$$${find}\: \\ $$$$\left.{a}\right){the}\:{workdone}\:{by}\:{the}\:{man}\:{on}\:{the} \\ $$$${box} \\ $$$$\left.{b}\right){workdone}\:{against}\:{friction} \\ $$

Question Number 34699    Answers: 0   Comments: 2

let f(x)=e^(−x^2 ) ∫_0 ^x e^t^2 dt 1) find a d.e verified by f 2) developpf at integr serie.

$${let}\:{f}\left({x}\right)={e}^{−{x}^{\mathrm{2}} } \:\int_{\mathrm{0}} ^{{x}} \:{e}^{{t}^{\mathrm{2}} } {dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{d}.{e}\:{verified}\:{by}\:{f} \\ $$$$\left.\mathrm{2}\right)\:{developpf}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 34698    Answers: 1   Comments: 2

let f(x)=e^x sinx .developp f at integr serie.

$${let}\:{f}\left({x}\right)={e}^{{x}} \:{sinx}\:\:.{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 34697    Answers: 0   Comments: 1

let f(x)=((ln(1+x))/(1+x)) developp f at integr serie

$${let}\:{f}\left({x}\right)=\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 34696    Answers: 0   Comments: 1

let f(x) =(√(2−x)) developp f at integr serie and give the radius of convergence.

$${let}\:{f}\left({x}\right)\:=\sqrt{\mathrm{2}−{x}} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:{and}\:{give}\:{the}\:{radius}\:{of} \\ $$$${convergence}. \\ $$

Question Number 34695    Answers: 0   Comments: 0

let give U_n = {(1/n) Π_(k=1) ^n (α+k)}^(1/n) find lim_(n→+∞) U_n ?.

$${let}\:{give}\:{U}_{{n}} =\:\left\{\frac{\mathrm{1}}{{n}}\:\prod_{{k}=\mathrm{1}} ^{{n}} \:\left(\alpha+{k}\right)\right\}^{\frac{\mathrm{1}}{{n}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \:?. \\ $$

Question Number 34694    Answers: 0   Comments: 0

find lim_(n→+∞) Σ_(k=n) ^(2n−1) (1/(2k+1)) .

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\sum_{{k}={n}} ^{\mathrm{2}{n}−\mathrm{1}} \:\:\frac{\mathrm{1}}{\mathrm{2}{k}+\mathrm{1}}\:. \\ $$

Question Number 34693    Answers: 0   Comments: 0

let S_n = (1/(n(√n))) Σ_(k=1) ^n [(√k) ] find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\:\frac{\mathrm{1}}{{n}\sqrt{{n}}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\left[\sqrt{{k}}\:\right] \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 34692    Answers: 0   Comments: 1

find lim_(n→+∞) (1/n^3 ) Σ_(k=1) ^n k^2 sin(((kπ)/n))

$${find}\:{lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}^{\mathrm{3}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{2}} \:{sin}\left(\frac{{k}\pi}{{n}}\right) \\ $$

Question Number 34691    Answers: 0   Comments: 0

calculate Σ_(n=3) ^∞ ((2n−1)/(n^3 −4n)) .

$${calculate}\:\:\sum_{{n}=\mathrm{3}} ^{\infty} \:\:\:\frac{\mathrm{2}{n}−\mathrm{1}}{{n}^{\mathrm{3}} \:−\mathrm{4}{n}}\:. \\ $$

Question Number 34690    Answers: 0   Comments: 0

let U_n =(1/(n!)) ∫_0 ^1 (arcsinx)^n dx calculate lim_(n→+∞) U_n .

$${let}\:{U}_{{n}} =\frac{\mathrm{1}}{{n}!}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left({arcsinx}\right)^{{n}} {dx} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:. \\ $$

Question Number 34689    Answers: 0   Comments: 1

prove that Σ_(k=1) ^n sin((k/n^2 )) =(1/2) +(1/(2n)) +o((1/n))

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}{n}}\:+{o}\left(\frac{\mathrm{1}}{{n}}\right) \\ $$

Question Number 34688    Answers: 0   Comments: 1

cslculate Σ_(n=2) ^∞ ln(1+(((−1)^n )/n))

$${cslculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\right) \\ $$

Question Number 34687    Answers: 0   Comments: 0

calculate Σ_(n=2) ^∞ ((1/(√(n−1))) + (1/(√(n+1))) −(2/(√n)))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\left(\frac{\mathrm{1}}{\sqrt{{n}−\mathrm{1}}}\:+\:\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}}\:−\frac{\mathrm{2}}{\sqrt{{n}}}\right) \\ $$

Question Number 34686    Answers: 0   Comments: 0

decompose F(x) = (((2n)!)/((x^2 −1)(x^2 −2)....(x^2 −n)))

$${decompose}\:{F}\left({x}\right)\:\:=\:\frac{\left(\mathrm{2}{n}\right)!}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} \:−\mathrm{2}\right)....\left({x}^{\mathrm{2}} \:−{n}\right)} \\ $$

Question Number 34685    Answers: 0   Comments: 0

decompose the fraction F(x)= (1/((x+2)( x^n −1))) with n ∈ N^★

$${decompose}\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}+\mathrm{2}\right)\left(\:{x}^{{n}} \:\:−\mathrm{1}\right)}\:\:{with}\:{n}\:\in\:{N}^{\bigstar} \\ $$

Question Number 34684    Answers: 0   Comments: 0

let U_n = (π/4) −Σ_(k=0) ^n (((−1)^k )/(2k+1)) calcilate Σ_(n=0) ^∞ U_n

$${let}\:{U}_{{n}} =\:\frac{\pi}{\mathrm{4}}\:−\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$${calcilate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 34683    Answers: 0   Comments: 0

find?the nature of Σ_(n=0) ^∞ sin{π(2+(√3) )^n }

$${find}?{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left\{\pi\left(\mathrm{2}+\sqrt{\mathrm{3}}\:\right)^{{n}} \right\} \\ $$

Question Number 34682    Answers: 0   Comments: 0

calculate Σ_(n=0) ^∞ ln(cos((a/2^n )))

$${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:{ln}\left({cos}\left(\frac{{a}}{\mathrm{2}^{{n}} }\right)\right) \\ $$

Question Number 34681    Answers: 0   Comments: 0

calculate lim_(x→0) { ((1+tanx)/(1+thx))}^(1/(sinx)) .

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\left\{\:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{thx}}\right\}^{\frac{\mathrm{1}}{{sinx}}} . \\ $$

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