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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)

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

Question Number 63164 Answers: 0 Comments: 0

let S_n =Σ_(k=1) ^n (((−1)^k )/k) and H_n =Σ_(k=1) ^n (1/k) calculate S_n interms of H_n 2)find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:\:\:\:\:\:{and}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$${calculate}\:{S}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

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