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Question Number 33915 Answers: 2 Comments: 3

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) find Γ^((n)) (x) with n∈ N^★ 2) calculate Γ(n +(3/2)) for n integr.

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma^{\left({n}\right)} \left({x}\right)\:{with}\:{n}\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\:+\frac{\mathrm{3}}{\mathrm{2}}\right)\:{for}\:{n}\:{integr}. \\ $$

Question Number 33912 Answers: 0 Comments: 3

Question Number 33907 Answers: 0 Comments: 1

Question Number 33899 Answers: 2 Comments: 0

express ((4t^2 −28)/(t^4 +t^2 −6)) as a partial fraction.

$$\boldsymbol{\mathrm{express}}\:\frac{\mathrm{4}\boldsymbol{\mathrm{t}}^{\mathrm{2}} −\mathrm{28}}{\boldsymbol{\mathrm{t}}^{\mathrm{4}} +\boldsymbol{\mathrm{t}}^{\mathrm{2}} −\mathrm{6}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{partial}}\:\boldsymbol{\mathrm{fraction}}. \\ $$

Question Number 33897 Answers: 2 Comments: 0

let consider ψ(x)=((Γ^′ (x))/(Γ(x))) 1) prove that ∀ a>0 ∫_0 ^1 ψ(a+x)dx=ln(a) 2) prove that ∀ n∈ N^★ ∫_0 ^1 ψ(x)sin(2πnx)dx=−(π/2)

$${let}\:{consider}\:\psi\left({x}\right)=\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \psi\left({a}+{x}\right){dx}={ln}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\in\:{N}^{\bigstar} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\psi\left({x}\right){sin}\left(\mathrm{2}\pi{nx}\right){dx}=−\frac{\pi}{\mathrm{2}} \\ $$

Question Number 33896 Answers: 3 Comments: 0

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt 1) find Γ(x+1) interms of Γ(x) with x>0 2)calculate Γ(n) for n ∈ N^★ 3)calculate Γ((3/2)) .

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma\left({x}+\mathrm{1}\right)\:{interms}\:{of}\:\Gamma\left({x}\right)\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\Gamma\left({n}\right)\:{for}\:{n}\:\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{3}\right){calculate}\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:. \\ $$

Question Number 33895 Answers: 3 Comments: 0

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) prove that Γ(x)Γ(1−x)= (π/(sin(πx))) 2) find the value of ∫_0 ^∞ e^(−x^2 ) dx .

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right)=\:\frac{\pi}{{sin}\left(\pi{x}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 33894 Answers: 0 Comments: 1

1)let f R→C 2π periodic even /f(x)=x ∀ x∈[0,π[ developp f at fourier serie 2) calculate Σ_(p=0) ^∞ (1/((2p+1)^2 )) .

$$\left.\mathrm{1}\right){let}\:{f}\:\:{R}\rightarrow{C}\:\:\mathrm{2}\pi\:{periodic}\:{even}\:\:/{f}\left({x}\right)={x}\: \\ $$$$\forall\:{x}\in\left[\mathrm{0},\pi\left[\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33892 Answers: 0 Comments: 1

prove that Σ_(n=1) ^∞ (H_n /n^2 ) =2 ξ(3) with ξ(x) =Σ_(n=1) ^∞ (1/n^x ) and x>1.

$${prove}\:{that}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{H}_{{n}} }{{n}^{\mathrm{2}} }\:=\mathrm{2}\:\xi\left(\mathrm{3}\right)\:{with} \\ $$$$\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{and}\:{x}>\mathrm{1}. \\ $$

Question Number 33891 Answers: 0 Comments: 1

let a∈C and ∣a∣<1 prove that the function f(x)= Σ_(n=0) ^(+∞) (a^n /(x+n)) is?developpable at point 1 and the radius is r=1.

$${let}\:{a}\in{C}\:{and}\:\mid{a}\mid<\mathrm{1}\:{prove}\:{that}\:{the}\:{function} \\ $$$${f}\left({x}\right)=\:\sum_{{n}=\mathrm{0}} ^{+\infty} \:\frac{{a}^{{n}} }{{x}+{n}}\:{is}?{developpable}\:{at}\:{point}\:\mathrm{1}\:{and} \\ $$$${the}\:{radius}\:{is}\:{r}=\mathrm{1}. \\ $$

Question Number 33890 Answers: 0 Comments: 0

find lim_(x→+∞) Σ_(n=1) ^∞ (1+(1/n))^n^2 (x^n /(n!)) .

$${find}\:{lim}_{{x}\rightarrow+\infty} \:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}^{\mathrm{2}} } \:\frac{{x}^{{n}} }{{n}!}\:. \\ $$

Question Number 33889 Answers: 0 Comments: 0

prove that ∫_0 ^1 (dx/(√(1−x^4 ))) = Σ_(n=0) ^∞ (C_(2n) ^n /(4^n (4n+1))) .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{C}_{\mathrm{2}{n}} ^{{n}} }{\mathrm{4}^{{n}} \left(\mathrm{4}{n}+\mathrm{1}\right)}\:. \\ $$

Question Number 33888 Answers: 0 Comments: 0

developp at integr serie f(x)= ∫_0 ^(π/2) (dt/(√(1−x^2 sin^2 t))) . with ∣x∣<1 .

$${developp}\:{at}\:{integr}\:{serie}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}}\:. \\ $$$${with}\:\mid{x}\mid<\mathrm{1}\:. \\ $$

Question Number 33887 Answers: 0 Comments: 0

find the value of Σ_(n=0) ^∞ (((−1)^n )/(4n+1)) .

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}+\mathrm{1}}\:. \\ $$

Question Number 33886 Answers: 0 Comments: 1

find the value of Σ_(n=0) ^∞ (((−1)^n )/(2n+3)).

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}+\mathrm{3}}. \\ $$

Question Number 33885 Answers: 0 Comments: 1

developp at integr serie f(x)= ∫_0 ^x sin(t^2 )dt .

$${developp}\:{at}\:{integr}\:{serie}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {sin}\left({t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 33884 Answers: 0 Comments: 1

let F(x)= ∫_0 ^(π/2) ((arctan(xtant))/(tant)) dt find a simple form of f(x) . 2) find the value of ∫_0 ^(π/2) ((arctan(2tant))/(tant))dt .

$${let}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{arctan}\left({xtant}\right)}{{tant}}\:{dt}\:{find}\:{a}\:{simple} \\ $$$${form}\:{of}\:{f}\left({x}\right)\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{arctan}\left(\mathrm{2}{tant}\right)}{{tant}}{dt}\:. \\ $$

Question Number 33883 Answers: 0 Comments: 1

find a simple form of f(x)=∫_0 ^(π/2) ln(1+xsin^2 t)dt with ∣x∣<1.

$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} {t}\right){dt} \\ $$$${with}\:\mid{x}\mid<\mathrm{1}. \\ $$

Question Number 33867 Answers: 1 Comments: 0

Given A(t) is an area bounded between y = x^2 + tx and x−axis, 0 < t < 2 Find the propability we choose t so (1/(48)) ≤ A(t) ≤ (1/(16))

$$\mathrm{Given}\:{A}\left({t}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{between}\: \\ $$$${y}\:=\:{x}^{\mathrm{2}} \:+\:{tx}\:\mathrm{and}\:\mathrm{x}−\mathrm{axis},\:\:\mathrm{0}\:<\:{t}\:<\:\mathrm{2} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{propability}\:\mathrm{we}\:\mathrm{choose}\:{t} \\ $$$$\mathrm{so}\:\frac{\mathrm{1}}{\mathrm{48}}\:\leqslant\:{A}\left({t}\right)\:\leqslant\:\frac{\mathrm{1}}{\mathrm{16}} \\ $$

Question Number 33865 Answers: 0 Comments: 2

evaluate lim_(x→∞) π (((aπ)^x )/(x!))

$${evaluate} \\ $$$$\:{li}\underset{{x}\rightarrow\infty} {{m}}\:\:\:\pi\:\frac{\left({a}\pi\right)^{{x}} }{{x}!} \\ $$

Question Number 33880 Answers: 1 Comments: 1

Question Number 33860 Answers: 0 Comments: 0

(√([x+3x]×4)) =( determinant (((2 4 5)),((3 5 7))))

$$\sqrt{\left[{x}+\mathrm{3}{x}\right]×\mathrm{4}}\:\:=\left(\begin{vmatrix}{\mathrm{2}\:\:\:\:\mathrm{4}\:\:\:\mathrm{5}}\\{\mathrm{3}\:\:\:\:\mathrm{5}\:\:\:\:\mathrm{7}}\end{vmatrix}\right) \\ $$

Question Number 33848 Answers: 0 Comments: 0

let w_n = (H_n ^2 /n) with H_n =Σ_(k=1) ^n (1/k) study the convergence of Σ_(n=1) ^∞ w_n x^n .

$${let}\:{w}_{{n}} =\:\frac{{H}_{{n}} ^{\mathrm{2}} }{{n}}\:\:\:{with}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{w}_{{n}} {x}^{{n}} \:\:. \\ $$

Question Number 33847 Answers: 0 Comments: 1

let give a sequence of real numbets positif (a_i )_(1≤i≤n) 1) prove that (Σ_(i=1) ^n a_i )^2 ≤ n Σ_(i=1) ^n a_i ^2 2)let put H_n =Σ_(k=1) ^n (1/k) and w_n = (H_n ^2 /n) prove that the sequence w_n is convergent .

$$\:{let}\:{give}\:{a}\:{sequence}\:{of}\:{real}\:{numbets}\:{positif} \\ $$$$\left({a}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:{a}_{{i}} \right)^{\mathrm{2}} \leqslant\:{n}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{a}_{{i}} ^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){let}\:{put}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:\:{and}\:{w}_{{n}} =\:\frac{{H}_{{n}} ^{\mathrm{2}} }{{n}} \\ $$$${prove}\:{that}\:{the}\:{sequence}\:{w}_{{n}} \:{is}\:{convergent}\:. \\ $$

Question Number 33846 Answers: 0 Comments: 1

find radous of conbergence for theserie Σ_(n≥0) x^(n!) .

$${find}\:{radous}\:{of}\:{conbergence}\:{for}\:{theserie}\:\sum_{{n}\geqslant\mathrm{0}} {x}^{{n}!} .\: \\ $$

Question Number 33845 Answers: 0 Comments: 1

let I_n = ∫_0 ^1 ((arctan(1 +n))/(√(1+x^n ))) find lim_(n→+∞) I_n .

$${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left(\mathrm{1}\:+{n}\right)}{\sqrt{\mathrm{1}+{x}^{{n}} }}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{I}_{{n}} \:. \\ $$

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