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Question Number 63246 Answers: 0 Comments: 2

Arrange these digits: 1 1 2 2 3 3 4 4 So that the 1′s are four digit apart So that the 2′s are three digit apart So that the 3′s are two digit apart So that the 4′s are one digit apart

$$\mathrm{Arrange}\:\mathrm{these}\:\mathrm{digits}:\:\:\:\:\:\:\mathrm{1}\:\:\mathrm{1}\:\:\mathrm{2}\:\:\mathrm{2}\:\:\mathrm{3}\:\:\mathrm{3}\:\:\mathrm{4}\:\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{1}'\mathrm{s}\:\mathrm{are}\:\mathrm{four}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{2}'\mathrm{s}\:\mathrm{are}\:\mathrm{three}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{3}'\mathrm{s}\:\mathrm{are}\:\mathrm{two}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{4}'\mathrm{s}\:\mathrm{are}\:\mathrm{one}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$ \\ $$

Question Number 63256 Answers: 0 Comments: 3

Question Number 63233 Answers: 0 Comments: 4

Question Number 63232 Answers: 0 Comments: 2

let B(x,y) =∫_0 ^1 (1−t)^(x−1) t^(y−1) dt 1) study the convergence of B(x,y) 1) prove that B(x,y)=B(y,x) prove that B(x,y) =∫_0 ^∞ (t^(x−1) /((1+t)^(x+y) )) dt 2) prove that B(x,y) =((Γ(x).Γ(y))/(Γ(x+y))) 3) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) for allx ∈]0,1[

$${let}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{x}−\mathrm{1}} {t}^{{y}−\mathrm{1}} \:{dt} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convergence}\:{of}\:{B}\left({x},{y}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{B}\left({x},{y}\right)={B}\left({y},{x}\right) \\ $$$${prove}\:{that}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{x}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{{x}+{y}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{B}\left({x},{y}\right)\:=\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:{for}\:{allx}\:\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$

Question Number 63225 Answers: 0 Comments: 0

Question Number 63645 Answers: 0 Comments: 4

n integr natural prove that 5 divide n^5 −n

$${n}\:{integr}\:{natural}\:{prove}\:{that}\:\mathrm{5}\:{divide}\:{n}^{\mathrm{5}} −{n} \\ $$

Question Number 63251 Answers: 0 Comments: 0

∫_( 0) ^( (π/2)) sin^(−1) (m cosθ) dθ

$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{m}\:\mathrm{cos}\theta\right)\:\mathrm{d}\theta \\ $$

Question Number 63215 Answers: 0 Comments: 1

calculate lim_(n→+∞) {n (1+(1/n))^n −en}

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \left\{{n}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} −{en}\right\} \\ $$

Question Number 63214 Answers: 0 Comments: 1

calculate ∫_0 ^∞ x e^(−(x^2 /a^2 )) sin(bx)dx with a>0 and b>0

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:{x}\:{e}^{−\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }} \:\:{sin}\left({bx}\right){dx}\:\:{with}\:\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 63267 Answers: 0 Comments: 3

lim_(n→∞) (((n^3 + 1)/(n^3 − 1)))^(2n − n^3 )

$$\:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\left(\frac{\mathrm{n}^{\mathrm{3}} \:+\:\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \:−\:\mathrm{1}}\right)^{\mathrm{2n}\:−\:\mathrm{n}^{\mathrm{3}} } \\ $$

Question Number 63206 Answers: 1 Comments: 1

Question Number 63203 Answers: 0 Comments: 5

Question Number 63268 Answers: 0 Comments: 0

Question Number 63194 Answers: 1 Comments: 0

Question Number 63190 Answers: 0 Comments: 3

Test its convergence: Σ_(n = 1) ^∞ (1/(n^3 sin^2 n))

$$\mathrm{Test}\:\mathrm{its}\:\mathrm{convergence}:\:\:\:\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \:\mathrm{sin}^{\mathrm{2}} \mathrm{n}} \\ $$

Question Number 63292 Answers: 0 Comments: 4

∫_0 ^1 ∫_0 ^1 (dy/(1+y(x^2 −x))) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dy}}{\mathrm{1}+{y}\left({x}^{\mathrm{2}} −{x}\right)}\:{dx} \\ $$

Question Number 63291 Answers: 0 Comments: 3

find some of all real x such that ((4x^2 +15x+17)/(x^2 +4x+12)) = ((5x^2 +16x+18)/(2x^2 +5x+13))

$${find}\:{some}\:{of}\:{all}\:{real}\:{x}\:{such}\:{that} \\ $$$$ \\ $$$$\frac{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{15}{x}+\mathrm{17}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{12}}\:=\:\frac{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{16}{x}+\mathrm{18}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{13}} \\ $$

Question Number 63290 Answers: 0 Comments: 0

Question Number 63289 Answers: 0 Comments: 0

Question Number 63288 Answers: 1 Comments: 0

Question Number 63218 Answers: 0 Comments: 8

Q.63108 (A check) eq. of ellipse (x^2 /4)+y^2 =1 Inscribed equilateral △ABC of side s=((16(√6))/(√(365))) Do these points satisfy for A, B, C ? A((4/(√(365))), ((19)/(√(365)))) ; B[−(((20+8(√3)))/(√(365))), ((8(√3)−5)/(√(365)))] C[−(((20−8(√3)))/(√(365))) , −(((5+8(√3)))/(√(365)))] θ=45°

$${Q}.\mathrm{63108}\:\:\:\left({A}\:{check}\right) \\ $$$${eq}.\:{of}\:{ellipse} \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{4}}+{y}^{\mathrm{2}} =\mathrm{1} \\ $$$${Inscribed}\:{equilateral}\:\bigtriangleup{ABC} \\ $$$${of}\:{side}\:\boldsymbol{{s}}=\frac{\mathrm{16}\sqrt{\mathrm{6}}}{\sqrt{\mathrm{365}}} \\ $$$${Do}\:{these}\:{points}\:{satisfy}\:{for} \\ $$$${A},\:{B},\:{C}\:? \\ $$$${A}\left(\frac{\mathrm{4}}{\sqrt{\mathrm{365}}},\:\frac{\mathrm{19}}{\sqrt{\mathrm{365}}}\right)\:\:\:;\:\: \\ $$$${B}\left[−\frac{\left(\mathrm{20}+\mathrm{8}\sqrt{\mathrm{3}}\right)}{\sqrt{\mathrm{365}}},\:\frac{\mathrm{8}\sqrt{\mathrm{3}}−\mathrm{5}}{\sqrt{\mathrm{365}}}\right] \\ $$$${C}\left[−\frac{\left(\mathrm{20}−\mathrm{8}\sqrt{\mathrm{3}}\right)}{\sqrt{\mathrm{365}}}\:,\:−\frac{\left(\mathrm{5}+\mathrm{8}\sqrt{\mathrm{3}}\right)}{\sqrt{\mathrm{365}}}\right] \\ $$$$\:\theta=\mathrm{45}° \\ $$

Question Number 63178 Answers: 2 Comments: 1

Question Number 63176 Answers: 1 Comments: 1

Question Number 63651 Answers: 0 Comments: 0

let S_n (x)=Σ_(k=0) ^n e^(−k) sin(k^2 x) 1) determine 2 sequence U_n (x) and V_n (x) wich verify U_n ≤ S_n ≤ V_n 2) let S =lim_(n→+∞) S(x) study the convergence of S.

$${let}\:{S}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{e}^{−{k}} {sin}\left({k}^{\mathrm{2}} {x}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:\mathrm{2}\:{sequence}\:\:{U}_{{n}} \left({x}\right)\:{and}\:{V}_{{n}} \left({x}\right)\:{wich}\:{verify}\:{U}_{{n}} \leqslant\:{S}_{{n}} \leqslant\:{V}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}\:={lim}_{{n}\rightarrow+\infty} \:{S}\left({x}\right)\:\:{study}\:{the}\:{convergence}\:{of}\:{S}. \\ $$

Question Number 63175 Answers: 0 Comments: 2

solve for x x^x^x = 16 x = 2, but how to use Lambert W function

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \:=\:\:\mathrm{16} \\ $$$$\mathrm{x}\:=\:\mathrm{2},\:\:\:\:\:\mathrm{but}\:\mathrm{how}\:\mathrm{to}\:\mathrm{use}\:\mathrm{Lambert}\:\mathrm{W}\:\mathrm{function} \\ $$

Question Number 63165 Answers: 0 Comments: 0

let W_n =Σ_(k=0) ^n (1/(3k+1)) determine W_n interms of H_n H_n =Σ_(k=1) ^n (1/k)

$${let}\:{W}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{1}}\:\:\:{determine}\:{W}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

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