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let give x>0 1) prove that ∫_0 ^1 (dt/(1+t^x ))= Σ_(n=0) ^∞ (((−1)^n )/(nx+1)) 2) find Σ_(n=0) ^∞ (((−1)^n )/(n+1)) and Σ_(n=0) ^∞ (((−1)^n )/(2n+1)) 3) find Σ_(n=1) ^∞ (((−1)^n )/(3n+1)) .

$${let}\:{give}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{{x}} }=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{nx}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}\:\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}+\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{3}{n}+\mathrm{1}}\:. \\ $$

Question Number 29976 Answers: 0 Comments: 0

prove that ln(Γ(x))= −lnx −γx +Σ_(n=1) ^∞ ( (x/n) −ln( 1+(x/n))) with x>0

$${prove}\:{that} \\ $$$${ln}\left(\Gamma\left({x}\right)\right)=\:−{lnx}\:−\gamma{x}\:\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\left(\:\:\frac{{x}}{{n}}\:\:−{ln}\left(\:\mathrm{1}+\frac{{x}}{{n}}\right)\right)\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 29975 Answers: 0 Comments: 2

let give 0<α<1 1) prove that π coth(πα) −(1/α) = Σ_(n=1) ^∞ ((2α)/(α^2 +n^2 )). 2)by integration on[0,1] find Π_(n=1) ^∞ (1+(1/n^2 )).

$$\:{let}\:{give}\:\mathrm{0}<\alpha<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\pi\:{coth}\left(\pi\alpha\right)\:−\frac{\mathrm{1}}{\alpha}\:=\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{2}\alpha}{\alpha^{\mathrm{2}} \:+{n}^{\mathrm{2}} }. \\ $$$$\left.\mathrm{2}\right){by}\:{integration}\:{on}\left[\mathrm{0},\mathrm{1}\right]\:{find}\:\prod_{{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right). \\ $$

Question Number 29974 Answers: 1 Comments: 0

(√(5=?))

$$\sqrt{\mathrm{5}=?} \\ $$

Question Number 29973 Answers: 0 Comments: 1

find Σ_(n=1) ^∞ ((sin(nα))/n) x^n with −1<x<1.

$${find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({n}\alpha\right)}{{n}}\:{x}^{{n}} \:{with}\:\:−\mathrm{1}<{x}<\mathrm{1}. \\ $$

Question Number 29972 Answers: 0 Comments: 1

let give ∣x∣<1 find ∫_0 ^(π/2) (dθ/(√(1−x^2 cos^2 θ))) .

$${let}\:{give}\:\mid{x}\mid<\mathrm{1}\:{find}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{d}\theta}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} {cos}^{\mathrm{2}} \theta}}\:. \\ $$

Question Number 29971 Answers: 0 Comments: 2

find J(x)= ∫_0 ^∞ (dt/(x+e^t )) ?.

$${find}\:{J}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{x}+{e}^{{t}} }\:\:\:\:?. \\ $$

Question Number 29970 Answers: 0 Comments: 1

a>0 and b>0 if (1/((1−ax)(1−bx)))=Σ_n a_n x^n find the sequence a_n .

$${a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:\:{if}\:\:\:\frac{\mathrm{1}}{\left(\mathrm{1}−{ax}\right)\left(\mathrm{1}−{bx}\right)}=\sum_{{n}} \:\:{a}_{{n}} \:{x}^{{n}} \\ $$$${find}\:{the}\:{sequence}\:{a}_{{n}} . \\ $$

Question Number 30000 Answers: 1 Comments: 1

If cos α = sin β sin φ=sin γ cos ψ cos β = sin γ sin ψ =sin α cos θ cos γ = sin α sin θ =sin β cos φ then find cos α, cos β , cos γ briefly and if possible linearly in terms of only sin θ, cos θ, sin φ, cos φ, sin ψ, cos ψ .

$${If}\:\mathrm{cos}\:\alpha\:=\:\mathrm{sin}\:\beta\:\mathrm{sin}\:\phi=\mathrm{sin}\:\gamma\:\mathrm{cos}\:\psi \\ $$$$\:\:\:\:\:\:\mathrm{cos}\:\beta\:=\:\mathrm{sin}\:\gamma\:\mathrm{sin}\:\psi\:=\mathrm{sin}\:\alpha\:\mathrm{cos}\:\theta \\ $$$$\:\:\:\:\:\:\mathrm{cos}\:\gamma\:=\:\mathrm{sin}\:\alpha\:\mathrm{sin}\:\theta\:=\mathrm{sin}\:\beta\:\mathrm{cos}\:\phi \\ $$$${then}\:{find}\:\:\mathrm{cos}\:\alpha,\:\mathrm{cos}\:\beta\:,\:\mathrm{cos}\:\gamma\:\:\: \\ $$$${briefly}\:{and}\:{if}\:{possible}\:{linearly} \\ $$$${in}\:{terms}\:{of}\:{only}\:\mathrm{sin}\:\theta,\:\mathrm{cos}\:\theta, \\ $$$$\mathrm{sin}\:\phi,\:\mathrm{cos}\:\phi,\:\mathrm{sin}\:\psi,\:\mathrm{cos}\:\psi\:. \\ $$

Question Number 29960 Answers: 1 Comments: 0