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Question Number 163043    Answers: 0   Comments: 0

lim_(x→−∞) Σ_(n=1) ^∞ (x^n /n^n )=?

$$\underset{\mathrm{x}\rightarrow−\infty} {\mathrm{lim}}\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{n}^{\mathrm{n}} }=? \\ $$

Question Number 163042    Answers: 0   Comments: 0

lim_(n→∞) (√n)∫_(−∞) ^(+∞) ((cos x)/((1+x^2 )^n ))dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{\mathrm{n}}\int_{−\infty} ^{+\infty} \frac{\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} }\mathrm{dx}=? \\ $$

Question Number 163041    Answers: 0   Comments: 0

lim_(n→∞) (√n)(∫_0 ^1 ((1−(1−x^2 )^n )/( (√n)x^2 ))dx−(√π))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{\mathrm{n}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} }{\:\sqrt{\mathrm{n}}\mathrm{x}^{\mathrm{2}} }\mathrm{dx}−\sqrt{\pi}\right)=? \\ $$

Question Number 163040    Answers: 0   Comments: 0

lim_(n→∞) ((n!)/n^n )(Σ_(k=0) ^n (n^k /(k!))−Σ_(k=n+1) ^∞ (n^k /(k!)))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{n}!}{\mathrm{n}^{\mathrm{n}} }\left(\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}^{\mathrm{k}} }{\mathrm{k}!}−\underset{\mathrm{k}=\mathrm{n}+\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{n}^{\mathrm{k}} }{\mathrm{k}!}\right)=? \\ $$

Question Number 163039    Answers: 0   Comments: 0

lim_(n→∞) (√n)∫_0 ^1 x∙sin^(2n) (2πx)dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{\mathrm{n}}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}\centerdot\mathrm{sin}^{\mathrm{2n}} \left(\mathrm{2}\pi\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 163072    Answers: 1   Comments: 0

∫((sinx+sin3x+sin5x+sin7x)/(cosx+cos3x+cos5x+cos7x)) dx

$$\int\frac{\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{sin}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}\mathrm{7}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{cosx}}+\boldsymbol{\mathrm{cos}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{7}\boldsymbol{\mathrm{x}}}\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 163033    Answers: 1   Comments: 0

Ω= ∫_0 ^( ∞) (( x − sin (x ))/x^( 3) )dx −−− solution−−− Ω=^(I.B.P) [ ((−1)/(2 x^( 2) )) (x−sin(x))]_0 ^∞ +(1/2) ∫_0 ^( ∞) ((1−cos (x))/x^( 2) )dx = (1/2) ∫_0 ^( ∞) (( 2sin^( 2) ((x/2)))/x^( 2) )dx=∫_0 ^( ∞) ((sin^( 2) ((x/2)))/x^( 2) )dx =^((x/2) = α) (1/2)∫_0 ^( ∞) ((sin^( 2) ( α))/α^( 2) ) dα = (1/2) [((−1)/α) sin^( 2) (α)]_0 ^∞ +(1/2)∫_0 ^( ∞) ((sin(2α))/α)dα =^(2α=ϕ) (1/2) ∫_0 ^( ∞) (( sin(ϕ ))/ϕ) dϕ =(π/4) −− Ω= (π/4) −−−

$$ \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}\:−\:{sin}\:\left({x}\:\right)}{{x}^{\:\mathrm{3}} }{dx} \\ $$$$−−−\:{solution}−−− \\ $$$$\:\:\:\:\:\Omega\overset{\mathscr{I}.\mathscr{B}.\mathscr{P}} {=}\:\left[\:\frac{−\mathrm{1}}{\mathrm{2}\:{x}^{\:\mathrm{2}} }\:\left({x}−{sin}\left({x}\right)\right)\right]_{\mathrm{0}} ^{\infty} +\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}−{cos}\:\left({x}\right)}{{x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{2}{sin}^{\:\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{{x}^{\:\mathrm{2}} }{dx}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\:\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{{x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\overset{\frac{{x}}{\mathrm{2}}\:=\:\alpha} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\:\mathrm{2}} \left(\:\alpha\right)}{\alpha^{\:\mathrm{2}} }\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\left[\frac{−\mathrm{1}}{\alpha}\:{sin}^{\:\mathrm{2}} \left(\alpha\right)\right]_{\mathrm{0}} ^{\infty} +\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{2}\alpha\right)}{\alpha}{d}\alpha \\ $$$$\:\:\:\:\:\:\:\:\overset{\mathrm{2}\alpha=\varphi} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left(\varphi\:\right)}{\varphi}\:{d}\varphi\:=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−\:\:\:\:\:\Omega=\:\frac{\pi}{\mathrm{4}}\:\:−−− \\ $$$$ \\ $$$$ \\ $$

Question Number 163024    Answers: 0   Comments: 2

calcul en fonction de n Σ_(k=0) ^(2n) (_k ^(2n) )^(n−2k)

$${calcul}\:{en}\:{fonction}\:{de}\:{n} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}{n}} {\sum}}\left(_{{k}} ^{\mathrm{2}{n}} \right)^{{n}−\mathrm{2}{k}} \\ $$

Question Number 163020    Answers: 0   Comments: 2

cos^(2n) (x)+sin^(2n) (x)=((4^n +1)/5^n ) prove that

$$\boldsymbol{\mathrm{cos}}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \left(\boldsymbol{\mathrm{x}}\right)+\boldsymbol{\mathrm{sin}}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{4}^{\boldsymbol{\mathrm{n}}} +\mathrm{1}}{\mathrm{5}^{\boldsymbol{\mathrm{n}}} } \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$

Question Number 163014    Answers: 2   Comments: 0

given (a/b)=(c/d), find an expression for (a/(a+b))

$${given}\:\frac{{a}}{{b}}=\frac{{c}}{{d}},\:{find}\:{an}\:{expression} \\ $$$${for}\:\:\frac{{a}}{{a}+{b}} \\ $$

Question Number 163028    Answers: 1   Comments: 0

Question Number 163008    Answers: 2   Comments: 0

Calculate ∫ (((2−4sin x cos x)(1+sin 2x))/(sin^4 2x+64 cos^4 2x)) dx

$$\:{Calculate}\: \\ $$$$\:\:\:\int\:\frac{\left(\mathrm{2}−\mathrm{4sin}\:{x}\:\mathrm{cos}\:{x}\right)\left(\mathrm{1}+\mathrm{sin}\:\mathrm{2}{x}\right)}{\mathrm{sin}\:^{\mathrm{4}} \mathrm{2}{x}+\mathrm{64}\:\mathrm{cos}\:^{\mathrm{4}} \mathrm{2}{x}}\:{dx}\: \\ $$$$ \\ $$

Question Number 163002    Answers: 0   Comments: 0

prove that i:Σ_(n=0) ^∞ (((−1 )^( n) )/((n +(1/2))cosh(n+(1/2))π)) =(π/4) ii: ∫_0 ^( 1) (( sin( π x ))/(x^( x) ( 1−x )^( 1−x) )) (dx/(1+x)) =(π/4) −−−

$$ \\ $$$$\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:{i}:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\:\right)^{\:{n}} }{\left({n}\:+\frac{\mathrm{1}}{\mathrm{2}}\right){cosh}\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\pi}\:=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:{ii}:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{sin}\left(\:\pi\:{x}\:\right)}{{x}^{\:{x}} \left(\:\mathrm{1}−{x}\:\right)^{\:\mathrm{1}−{x}} }\:\frac{{dx}}{\mathrm{1}+{x}}\:=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:−−− \\ $$

Question Number 163000    Answers: 1   Comments: 0

Question Number 162999    Answers: 0   Comments: 0

Question Number 162995    Answers: 1   Comments: 0

if a_k > 0 ; k = 1,5^(−) then prove that exists i,j∈1,5^(−) such that: 0 ≤ ((a_j - a_i )/(1 + a_i a_j )) ≤ (√2) - 1

$$\mathrm{if}\:\:\mathrm{a}_{\boldsymbol{\mathrm{k}}} \:>\:\mathrm{0}\:\:;\:\:\mathrm{k}\:=\:\overline {\mathrm{1},\mathrm{5}} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{exists}\:\:\boldsymbol{\mathrm{i}},\boldsymbol{\mathrm{j}}\in\overline {\mathrm{1},\mathrm{5}}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\mathrm{0}\:\leqslant\:\frac{\mathrm{a}_{\boldsymbol{\mathrm{j}}} \:-\:\mathrm{a}_{\boldsymbol{\mathrm{i}}} }{\mathrm{1}\:+\:\mathrm{a}_{\boldsymbol{\mathrm{i}}} \mathrm{a}_{\boldsymbol{\mathrm{j}}} }\:\leqslant\:\sqrt{\mathrm{2}}\:-\:\mathrm{1} \\ $$

Question Number 162994    Answers: 0   Comments: 0

Question Number 162993    Answers: 0   Comments: 0

Find: 𝛀 =∫_( 0) ^( (𝛑/2)) xcot(x)log(cos(x))dx

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\mathrm{xcot}\left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{cos}\left(\mathrm{x}\right)\right)\mathrm{dx} \\ $$

Question Number 162974    Answers: 0   Comments: 0

Question Number 162973    Answers: 0   Comments: 0

Question Number 162972    Answers: 0   Comments: 0

Question Number 162971    Answers: 0   Comments: 0

Question Number 162960    Answers: 1   Comments: 0

A ∈ M_( n×n) , A^( 3) = O^− Find , ( A−2I )^( −1) =?

$$ \\ $$$$\:\:\:\:\mathrm{A}\:\in\:\mathrm{M}_{\:{n}×{n}} \:\:,\:\:\:\mathrm{A}^{\:\mathrm{3}} =\:\overset{−} {\mathrm{O}}\:\: \\ $$$$\:\:\:\:\:\mathrm{Find}\:,\:\:\:\:\:\:\:\:\left(\:\mathrm{A}−\mathrm{2I}\:\right)^{\:−\mathrm{1}} =? \\ $$$$ \\ $$

Question Number 162959    Answers: 1   Comments: 5

solve the differential equation y = x + p^3

$${solve}\:{the}\:{differential}\:{equation}\:{y}\:=\:{x}\:+\:{p}^{\mathrm{3}} \\ $$

Question Number 162952    Answers: 0   Comments: 0

Question Number 162951    Answers: 1   Comments: 2

4men clear a farm for 8 days and are paid 24$ How long will 6 men take to clear the same farm if they are paid 360$ ?

$$\mathrm{4men}\:\mathrm{clear}\:\mathrm{a}\:\mathrm{farm}\:\mathrm{for}\:\mathrm{8}\:\mathrm{days}\:\mathrm{and}\:\mathrm{are}\:\mathrm{paid}\:\mathrm{24\$} \\ $$$$\mathrm{How}\:\mathrm{long}\:\mathrm{will}\:\mathrm{6}\:\mathrm{men}\:\mathrm{take}\:\mathrm{to}\:\mathrm{clear}\:\mathrm{the}\:\mathrm{same}\:\mathrm{farm} \\ $$$$\mathrm{if}\:\mathrm{they}\:\mathrm{are}\:\mathrm{paid}\:\mathrm{360\$}\:? \\ $$

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