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Question Number 33349 Answers: 0 Comments: 1

prove that ∫_0 ^∞ x(x−ln(e^x −1))dx=Σ_(n=1) ^∞ (1/n^3 )

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{x}\left({x}−{ln}\left({e}^{{x}} −\mathrm{1}\right)\right){dx}=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} } \\ $$

Question Number 33348 Answers: 0 Comments: 0

let S_n = Σ_(k=1) ^∞ (((−1)^(k−1) )/k) and T_n = Σ_(k=1) ^∞ (((−1)^(k−1) )/(2k−1)) 1) calculate lim S_n and lim T_n (n→∞) 2)prove that Σ(S_n −ln2) and Σ(T_n −(π/4))converges and find its sum

$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{and} \\ $$$$\underset{{n}} {{T}}\:=\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{\mathrm{2}{k}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}\:{S}_{{n}} \:\:{and}\:{lim}\:{T}_{{n}} \left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\Sigma\left({S}_{{n}} −{ln}\mathrm{2}\right)\:{and} \\ $$$$\Sigma\left({T}_{{n}} \:−\frac{\pi}{\mathrm{4}}\right){converges}\:{and}\:{find}\:{its}\:{sum} \\ $$$$ \\ $$

Question Number 33347 Answers: 0 Comments: 0

let f_n (x)= nx^2 e^(−x(√n)) , x≥0 1)study the simple convergence of Σ f_n (x{ 2) study the uniform convergence of Σ f_n (x).

$${let}\:{f}_{{n}} \left({x}\right)=\:{nx}^{\mathrm{2}} \:{e}^{−{x}\sqrt{{n}}} \:\:\:,\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{simple}\:{convergence}\:{of} \\ $$$$\Sigma\:{f}_{{n}} \left({x}\left\{\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{uniform}\:{convergence}\:{of} \\ $$$$\Sigma\:{f}_{{n}} \left({x}\right). \\ $$

Question Number 33346 Answers: 0 Comments: 0

let a and b from R /a<b f [a,b]→C continje prove that ∀n ∈N ∫_a ^b (Π_(k=0) ^(n−1) (x+k))f(x)dx=0 ⇒ f=0

$${let}\:{a}\:{and}\:{b}\:{from}\:{R}\:/{a}<{b}\:\:{f}\:\:\left[{a},{b}\right]\rightarrow{C}\:{continje} \\ $$$${prove}\:{that}\:\:\forall{n}\:\in{N}\:\:\int_{{a}} ^{{b}} \left(\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({x}+{k}\right)\right){f}\left({x}\right){dx}=\mathrm{0} \\ $$$$\Rightarrow\:{f}=\mathrm{0} \\ $$

Question Number 33345 Answers: 0 Comments: 0

for x∈]0,+∞[ let ψ(x) = ((Γ^′ (x))/(Γ(x))) 1)prove that ψ(x) =−(1/x) −γ +x Σ_(n=1) ^∞ (1/(n(x+n))) 2)ptove that γ =−Γ^′ (1) 3) prove that ∫_0 ^∞ e^(−x) ln(x)dx =−γ .

$$\left.{for}\:{x}\in\right]\mathrm{0},+\infty\left[\:{let}\:\psi\left({x}\right)\:=\:\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)}\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\psi\left({x}\right)\:=−\frac{\mathrm{1}}{{x}}\:−\gamma\:+{x}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}\left({x}+{n}\right)} \\ $$$$\left.\mathrm{2}\right){ptove}\:{that}\:\gamma\:=−\Gamma^{'} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {ln}\left({x}\right){dx}\:=−\gamma\:. \\ $$

Question Number 33344 Answers: 0 Comments: 1

prove that ∀ α ∈]0,+∞[ lim_(n→∞) ∫_0 ^n (1−(x/n))^n x^(α−1) dx =Γ(α) .

$$\left.{prove}\:{that}\:\:\forall\:\alpha\:\in\right]\mathrm{0},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \:{x}^{\alpha−\mathrm{1}} {dx}\:=\Gamma\left(\alpha\right)\:. \\ $$

Question Number 33343 Answers: 0 Comments: 2

prove that ∀ α ∈]1,+∞[ lim_(n→∞) ∫_0 ^n (1+(x/n))^n e^(−αx) dx = (1/(α−1)) .

$$\left.{prove}\:{that}\:\forall\:\alpha\:\in\right]\mathrm{1},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} \:{e}^{−\alpha{x}} {dx}\:=\:\frac{\mathrm{1}}{\alpha−\mathrm{1}}\:. \\ $$

Question Number 33342 Answers: 0 Comments: 4

let I_n = ∫_0 ^1 (1/x)(1−(1−(x/n))^n )dx and J_n = ∫_1 ^n (1/x)(1−(x/n))^n dx ,n integr not 0 1) prove that lim_ I_n =∫_0 ^1 ((1−e^(−x) )/x)dx lim J_n = ∫_0 ^1 (e^(−(1/x)) /x) dx (n→∞) 2) prove that ∀n∈ N^★ I_n −J_n = Σ_(k=1) ^n (1/k) −ln(n) 3) prove that ∫_0 ^1 ((1−e^(−x) −e^(−(1/x)) )/x) =γ

$${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{{x}}\left(\mathrm{1}−\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \right){dx}\:{and} \\ $$$${J}_{{n}} \:=\:\int_{\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}}\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {dx}\:\:,{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{lim}_{} \:{I}_{{n}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−{e}^{−{x}} }{{x}}{dx} \\ $$$${lim}\:{J}_{{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{−\frac{\mathrm{1}}{{x}}} }{{x}}\:{dx}\:\left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\forall{n}\in\:{N}^{\bigstar} \:\:\:{I}_{{n}} \:−{J}_{{n}} \:=\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:−{ln}\left({n}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\mathrm{1}−{e}^{−{x}} \:−{e}^{−\frac{\mathrm{1}}{{x}}} }{{x}}\:=\gamma \\ $$

Question Number 33341 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ (dx/((1+x^2 )(a^2 +x^2 ))) 2) find the value of A(θ) = ∫_0 ^∞ (dx/((1+x^2 )( x^2 +1 −sin^2 θ))) 0<θ<(π/2) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of} \\ $$$${A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:{x}^{\mathrm{2}} \:+\mathrm{1}\:−{sin}^{\mathrm{2}} \theta\right)} \\ $$$$\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}}\:. \\ $$

Question Number 33340 Answers: 0 Comments: 1

find a equivalent of A_n = ∫_0 ^n (√(1 +(1−(x/n))^n )) dx (n→∞)

$${find}\:{a}\:{equivalent}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:\sqrt{\mathrm{1}\:+\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} }\:\:{dx}\:\left({n}\rightarrow\infty\right) \\ $$

Question Number 33339 Answers: 0 Comments: 1

find lim_(n→∞) ∫_0 ^∞ e^(−x^2 ) sin^n xdx

$${find}\:\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {xdx}\: \\ $$

Question Number 33338 Answers: 0 Comments: 1

find lim_(n→∞) ∫_0 ^(+∞) tan^n xdx .

$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{+\infty} \:{tan}^{{n}} {xdx}\:.\: \\ $$

Question Number 33337 Answers: 0 Comments: 0

let α>0 prove that Σ_(k=1) ^n (1−α(k/n))_(n→∞) ^n → (e^(−α) /(1−e^(−α) ))

$${let}\:\alpha>\mathrm{0}\:{prove}\:{that} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:\left(\mathrm{1}−\alpha\frac{{k}}{{n}}\right)_{{n}\rightarrow\infty} ^{{n}} \rightarrow\:\frac{{e}^{−\alpha} }{\mathrm{1}−{e}^{−\alpha} } \\ $$

Question Number 33336 Answers: 0 Comments: 1

study the convergence of Σ_(n=1) ^∞ ln(1+(x/n^2 ))

$${study}\:{the}\:{convergence}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} {ln}\left(\mathrm{1}+\frac{{x}}{{n}^{\mathrm{2}} }\right) \\ $$

Question Number 33335 Answers: 0 Comments: 1

1) give ?D_(n−1) (o) for f(x)= (1/(x+2)) 2) drcompose inside R(x) the fraction F(x) = (1/(x^n (x+2)))

$$\left.\mathrm{1}\right)\:{give}\:?{D}_{{n}−\mathrm{1}} \left({o}\right)\:\:{for}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}+\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{drcompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}^{{n}} \left({x}+\mathrm{2}\right)} \\ $$

Question Number 33333 Answers: 0 Comments: 0

let hive I_n = ∫_0 ^(π/2) (sinx)^n dx prove that I_n ∼ (√(π/(2n))) (n→∞)

$${let}\:{hive}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({sinx}\right)^{{n}} \:{dx} \\ $$$${prove}\:{that}\:\:{I}_{{n}} \:\sim\:\:\sqrt{\frac{\pi}{\mathrm{2}{n}}}\:\left({n}\rightarrow\infty\right) \\ $$$$ \\ $$

Question Number 33334 Answers: 0 Comments: 3

decompose F(x)= (x^2 /(x^4 −1)) imside R(x) 2) find the value of ∫_2 ^(+∞) (x^2 /(x^4 −1)) .

$${decompose}\:{F}\left({x}\right)=\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} \:−\mathrm{1}}\:{imside}\:{R}\left({x}\right)\: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\:\int_{\mathrm{2}} ^{+\infty} \:\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} \:−\mathrm{1}}\:\:. \\ $$

Question Number 33331 Answers: 0 Comments: 1

calcilate ∫_0 ^1 (dx/((1+x^2 )^3 ))

$${calcilate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$

Question Number 33330 Answers: 2 Comments: 0

Question Number 33329 Answers: 0 Comments: 1

find ∫ (dx/(√(4x−x^2 ))) .

$${find}\:\:\int\:\:\:\frac{{dx}}{\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 33328 Answers: 0 Comments: 1

find ∫_(π/4) ^(4/π) (1+(1/x^2 ))arctanx dx

$${find}\:\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\mathrm{4}}{\pi}} \:\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx} \\ $$

Question Number 33327 Answers: 0 Comments: 1

find ∫_0 ^1 (dx/(3 +e^(−x) ))

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{3}\:+{e}^{−{x}} } \\ $$

Question Number 33323 Answers: 0 Comments: 1

Question related to Q#33217 If A_1 ,A_2 ,...A_n are n points with integer coordinates of a plane such that every triangle whose vertices are any three of the above points has its centroid with at least one non-integer coordinate. Find the maximum possible n. Recall that if P(x_1 ,y_1 ),Q(x_2 ,y_2 ),R(x_3 ,y_3 ) are three vertices then centroid G is (((x_1 +x_2 +x_3 )/3) , ((y_1 +y_2 +y_3 )/3))

$$\mathrm{Question}\:\mathrm{related}\:\mathrm{to}\:\mathrm{Q}#\mathrm{33217} \\ $$$$\mathrm{If}\:\mathrm{A}_{\mathrm{1}} ,\mathrm{A}_{\mathrm{2}} ,...\mathrm{A}_{\mathrm{n}} \:\mathrm{are}\:\mathrm{n}\:\mathrm{points}\:\mathrm{with}\:\mathrm{integer} \\ $$$$\mathrm{coordinates}\:\mathrm{of}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{such}\:\mathrm{that}\:\mathrm{every}\:\mathrm{triangle} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{are}\:\mathrm{any}\:\mathrm{three}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{points}\:\mathrm{has}\:\mathrm{its}\:\mathrm{centroid}\:\mathrm{with}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one} \\ $$$$\mathrm{non}-\mathrm{integer}\:\mathrm{coordinate}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{possible}\:\mathrm{n}.\:\:\: \\ $$$$\mathrm{Recall}\:\mathrm{that}\:\mathrm{if}\:\mathrm{P}\left(\mathrm{x}_{\mathrm{1}} ,\mathrm{y}_{\mathrm{1}} \right),\mathrm{Q}\left(\mathrm{x}_{\mathrm{2}} ,\mathrm{y}_{\mathrm{2}} \right),\mathrm{R}\left(\mathrm{x}_{\mathrm{3}} ,\mathrm{y}_{\mathrm{3}} \right) \\ $$$$\mathrm{are}\:\mathrm{three}\:\mathrm{vertices}\:\mathrm{then}\:\mathrm{centroid}\:\mathrm{G}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} }{\mathrm{3}}\:,\:\frac{\mathrm{y}_{\mathrm{1}} +\mathrm{y}_{\mathrm{2}} +\mathrm{y}_{\mathrm{3}} }{\mathrm{3}}\right) \\ $$

Question Number 33322 Answers: 0 Comments: 4

a) px^2 + 3x + q=0 has roots [((α+β)/(αβ))]× αβ find p and q if x^2 − 7x + 4 = 0 with real and distinct roots has same roots.. b)if α and β are roots of x^2 + kx +2k+8=0. a) find k if one root is twice the other.

$$\left.{a}\right)\:{px}^{\mathrm{2}} +\:\mathrm{3}{x}\:+\:{q}=\mathrm{0}\:{has}\:{roots}\: \\ $$$$\:\:\left[\frac{\alpha+\beta}{\alpha\beta}\right]×\:\alpha\beta \\ $$$${find}\:{p}\:{and}\:{q}\:{if}\: \\ $$$$\:\:{x}^{\mathrm{2}} −\:\mathrm{7}{x}\:+\:\mathrm{4}\:=\:\mathrm{0}\:{with}\:{real}\:{and}\: \\ $$$${distinct}\:{roots}\:{has}\:{same}\:{roots}.. \\ $$$$\left.{b}\right){if}\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\: \\ $$$$\:{x}^{\mathrm{2}} +\:{kx}\:+\mathrm{2}{k}+\mathrm{8}=\mathrm{0}. \\ $$$$\left.{a}\right)\:{find}\:{k}\:{if}\:{one}\:{root}\:{is}\:{twice}\:{the}\:{other}. \\ $$

Question Number 33317 Answers: 1 Comments: 0

find the radius of a circle wich inscribes an equalateral triangle with perimeter of 24cm

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{radius}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{circle}} \\ $$$$\boldsymbol{\mathrm{wich}}\:\boldsymbol{\mathrm{inscribes}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{equalateral}}\:\boldsymbol{\mathrm{triangle}} \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{perimeter}}\:\boldsymbol{\mathrm{of}}\:\mathrm{24}\boldsymbol{\mathrm{cm}} \\ $$

Question Number 33316 Answers: 1 Comments: 0

If ((f(2x+2y))/(f(2x−2y))) = ((sin (x+y))/(sin (x−y))) . Then find f(x) ?

$${If}\:\:\frac{{f}\left(\mathrm{2}{x}+\mathrm{2}{y}\right)}{{f}\left(\mathrm{2}{x}−\mathrm{2}{y}\right)}\:=\:\frac{\mathrm{sin}\:\left({x}+{y}\right)}{\mathrm{sin}\:\left({x}−{y}\right)}\:. \\ $$$${Then}\:{find}\:{f}\left({x}\right)\:? \\ $$

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