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Question Number 30089 Answers: 0 Comments: 0

prove the convergence or divergence of (((n − 1)/n))_(n = 1) ^∞

$$\mathrm{prove}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{or}\:\mathrm{divergence}\:\mathrm{of}\:\:\:\:\left(\frac{\mathrm{n}\:−\:\mathrm{1}}{\mathrm{n}}\right)_{\mathrm{n}\:=\:\mathrm{1}} ^{\infty} \\ $$

Question Number 30087 Answers: 3 Comments: 0

solve: cos3x.cos^3 x+sin 3x.sin^3 x=0.

$$\mathrm{solve}: \\ $$$$\mathrm{cos3}{x}.{cos}^{\mathrm{3}} {x}+\mathrm{sin}\:\mathrm{3}{x}.\mathrm{sin}\:^{\mathrm{3}} {x}=\mathrm{0}. \\ $$

Question Number 30079 Answers: 4 Comments: 1

Question Number 30054 Answers: 1 Comments: 0

Question Number 30049 Answers: 0 Comments: 1

find Σ_(n=1) ^∞ ((n(n+1))/3^n ) .

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{3}^{{n}} }\:. \\ $$

Question Number 30045 Answers: 1 Comments: 2

Question Number 30030 Answers: 1 Comments: 2

Question Number 30017 Answers: 1 Comments: 0

find the elements of the ellipse given the following equation 1.(x^2 /(25))+(y^2 /4)=1 2.x^2 +9y^2 =9

$$\mathrm{find}\:\mathrm{the}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ellipse}\:\mathrm{given}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation} \\ $$$$\mathrm{1}.\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{25}}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{4}}=\mathrm{1} \\ $$$$\mathrm{2}.\mathrm{x}^{\mathrm{2}} +\mathrm{9y}^{\mathrm{2}} =\mathrm{9} \\ $$

Question Number 30008 Answers: 0 Comments: 1

integrate w.r.t x ∫(e^x^2 )dx

$${integrate}\:{w}.{r}.{t}\:{x} \\ $$$$\int\left({e}^{{x}^{\mathrm{2}} } \right){dx} \\ $$

Question Number 30002 Answers: 0 Comments: 5

Question Number 29998 Answers: 0 Comments: 0

Question Number 29989 Answers: 1 Comments: 0

Question Number 29987 Answers: 0 Comments: 0

prove that Σ_(n=1_(n≠p) ) ^∞ (1/(n^2 −p^2 )) = (3/(4p^2 )) .

$${prove}\:{that}\:\:\sum_{{n}=\mathrm{1}_{{n}\neq{p}} } ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:−{p}^{\mathrm{2}} }\:\:=\:\:\frac{\mathrm{3}}{\mathrm{4}{p}^{\mathrm{2}} }\:. \\ $$

Question Number 29986 Answers: 1 Comments: 1

find Σ_(n=0) ^∞ ((n+1)/4^n ) .

$${find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{n}+\mathrm{1}}{\mathrm{4}^{{n}} }\:. \\ $$

Question Number 29985 Answers: 0 Comments: 0

prove that Σ_(p=1) ^∞ (a^p /(1−a^(2p) )) = Σ_(p=1) ^∞ (a^(2p−1) /(1−a^(2p−1) )) .

$${prove}\:{that}\:\:\:\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\:\:\:\:\:\frac{{a}^{{p}} }{\mathrm{1}−{a}^{\mathrm{2}{p}} }\:=\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{{a}^{\mathrm{2}{p}−\mathrm{1}} }{\mathrm{1}−{a}^{\mathrm{2}{p}−\mathrm{1}} }\:. \\ $$

Question Number 29984 Answers: 0 Comments: 0

prove that Σ_(n=1) ^∞ (H_n /(n!))==e Σ_(n=1) ^∞ (((1)^(n−1) )/(n (n!))) .

$${prove}\:{that}\:\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{H}_{{n}} }{{n}!}=={e}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}\right)^{\boldsymbol{{n}}−\mathrm{1}} }{\boldsymbol{{n}}\:\left(\boldsymbol{{n}}!\right)}\:. \\ $$

Question Number 29983 Answers: 0 Comments: 1

find radius andsum of Σ_(n=1) ^∞ ((n−1)/(n!)) x^n .

$${find}\:{radius}\:{andsum}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}−\mathrm{1}}{{n}!}\:{x}^{{n}} . \\ $$

Question Number 29982 Answers: 0 Comments: 0

let give f(x)=(√(x+(√(1+x^2 )))) developp f at integr series in point 0

$${let}\:{give}\:{f}\left({x}\right)=\sqrt{{x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}}\:\:\:\:{developp}\:{f}\:{at}\:{integr}\:{series} \\ $$$${in}\:{point}\:\mathrm{0} \\ $$

Question Number 29981 Answers: 0 Comments: 1

find radius and sum of Σ_(n=0) ^∞ (x^(2n) /(2n+1)) 2) find Σ_(n=0) ^∞ (1/((2n+1)9^n )) .

$${find}\:{radius}\:{and}\:{sum}\:{of}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{2}{n}} }{\mathrm{2}{n}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)\mathrm{9}^{{n}} }\:. \\ $$

Question Number 29980 Answers: 0 Comments: 0

prove that γ= Σ_(n=1) ^∞ ((1/n) −ln(1 +(1/n))) 2)show that γ= Σ_(k=2) ^∞ (((−1)^k )/k) ξ(k).

$${prove}\:{that}\:\gamma=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\left(\frac{\mathrm{1}}{{n}}\:\:−{ln}\left(\mathrm{1}\:+\frac{\mathrm{1}}{{n}}\right)\right) \\ $$$$\left.\mathrm{2}\right){show}\:{that}\:\gamma=\:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:\xi\left({k}\right). \\ $$

Question Number 29979 Answers: 0 Comments: 1

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

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

Question Number 29978 Answers: 0 Comments: 2

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.

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

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