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Question Number 32037 Answers: 0 Comments: 1

let u_n =cos(π(√(n^2 +n+1))) find nature of Σ u_n .

$${let}\:\:{u}_{{n}} ={cos}\left(\pi\sqrt{{n}^{\mathrm{2}} \:+{n}+\mathrm{1}}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} . \\ $$$$ \\ $$

Question Number 32036 Answers: 0 Comments: 0

nature of Σ u_n with u_n = (1/((ln(2))^2 +....+(ln(n))^2 )) .

$${nature}\:{of}\:\Sigma\:{u}_{{n}} \:\:{with}\:{u}_{{n}} =\:\:\:\frac{\mathrm{1}}{\left({ln}\left(\mathrm{2}\right)\right)^{\mathrm{2}} \:+....+\left({ln}\left({n}\right)\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 32034 Answers: 0 Comments: 0

let u_n = ∫_0 ^1 (dx/(1+x+...+x^n )) study the convergence of Σ u_n .

$${let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+...+{x}^{{n}} }\:\:{study}\:{the}\:{convergence}\:{of} \\ $$$$\Sigma\:{u}_{{n}} \:\:. \\ $$

Question Number 32033 Answers: 0 Comments: 0

let consider the sequence (u_n ) /u_0 ∈[0,1] and ∀n∈N u_(n+1) = u_n −u_n ^2 1) give a simple equivalent of u_n 2) find the nature of Σ u_n .

$${let}\:{consider}\:{the}\:{sequence}\:\:\left({u}_{{n}} \right)\:\:/{u}_{\mathrm{0}} \in\left[\mathrm{0},\mathrm{1}\right]\:{and} \\ $$$$\forall{n}\in{N}\:\:{u}_{{n}+\mathrm{1}} =\:{u}_{{n}} \:−{u}_{{n}} ^{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{simple}\:{equivalent}\:{of}\:\:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} . \\ $$

Question Number 32031 Answers: 0 Comments: 1

let f(a) = ∫_0 ^∞ e^(−ax) ln(x)dx with a>0 1) find f(a) 2) find ∫_0 ^∞ e^(−ax) (xlnx)dx 3) calculate ∫_0 ^∞ e^(−2x) (xlnx)dx .

$${let}\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} \left({xlnx}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}} \left({xlnx}\right){dx}\:\:. \\ $$

Question Number 32029 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−αx) ln(x) dx with α>0 .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\alpha{x}} {ln}\left({x}\right)\:{dx}\:\:{with}\:\:\alpha>\mathrm{0}\:. \\ $$

Question Number 32028 Answers: 1 Comments: 0

If ((2z_1 )/(3z_2 )) is a purely imaginary number, then find the value of ∣((z_1 −z_2 )/(z_1 +z_2 ))∣ .

$${If}\:\frac{\mathrm{2}{z}_{\mathrm{1}} }{\mathrm{3}{z}_{\mathrm{2}} }\:{is}\:{a}\:{purely}\:{imaginary}\:{number}, \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\mid\frac{{z}_{\mathrm{1}} −{z}_{\mathrm{2}} }{{z}_{\mathrm{1}} +{z}_{\mathrm{2}} }\mid\:. \\ $$

Question Number 32026 Answers: 0 Comments: 1

let α>0 prove that Σ_(n=0) ^∞ (((−1)^n )/(n+α)) =∫_0 ^1 (x^(α−1) /(1+x))dx .

$${let}\:\alpha>\mathrm{0}\:{prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\alpha}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\alpha−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:. \\ $$

Question Number 32025 Answers: 0 Comments: 2

calculate Σ_(n=0) ^∞ (((−1)^n )/(4n+1)) .

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

Question Number 32008 Answers: 1 Comments: 3

lim_(n→∞) [((1/n))^n +((2/n))^n +..+((n/n))^n ]=...

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left[\left(\frac{\mathrm{1}}{{n}}\right)^{{n}} +\left(\frac{\mathrm{2}}{{n}}\right)^{{n}} +..+\left(\frac{{n}}{{n}}\right)^{{n}} \right]=... \\ $$

Question Number 32002 Answers: 1 Comments: 0

If z^3 =z^ prove then ∣z∣=1.

$${If}\:\:\boldsymbol{{z}}^{\mathrm{3}} =\bar {\boldsymbol{{z}}}\:{prove}\: \\ $$$${then}\:\mid\boldsymbol{{z}}\mid=\mathrm{1}. \\ $$

Question Number 31991 Answers: 0 Comments: 1

g_n =(√(g_(n−1) +g_(n−2) )) g_1 =1 g_2 =3 g_n =..

$${g}_{{n}} =\sqrt{{g}_{{n}−\mathrm{1}} +{g}_{{n}−\mathrm{2}} } \\ $$$${g}_{\mathrm{1}} =\mathrm{1} \\ $$$${g}_{\mathrm{2}} =\mathrm{3} \\ $$$${g}_{{n}} =.. \\ $$

Question Number 31990 Answers: 1 Comments: 3

a_n =2a_(n−1) +3a_(n−2) a_0 =1 a_1 =2 a_n =...

$${a}_{{n}} =\mathrm{2}{a}_{{n}−\mathrm{1}} +\mathrm{3}{a}_{{n}−\mathrm{2}} \\ $$$${a}_{\mathrm{0}} =\mathrm{1} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{{n}} =... \\ $$

Question Number 31984 Answers: 0 Comments: 1

study the covergence of Σ u_n with u_n =^n (√(n/(n+1))) −1 .

$${study}\:{the}\:{covergence}\:{of}\:\:\Sigma\:{u}_{{n}} \:\:{with} \\ $$$${u}_{{n}} =^{{n}} \sqrt{\frac{{n}}{{n}+\mathrm{1}}}\:−\mathrm{1}\:\:\:. \\ $$

Question Number 31983 Answers: 0 Comments: 2

calculate Σ_(n=0) ^∞ ((n^2 −2)/(n!)) .

$${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} \:−\mathrm{2}}{{n}!}\:\:. \\ $$

Question Number 31982 Answers: 0 Comments: 2

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

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

Question Number 31981 Answers: 0 Comments: 1

find the nature of Σ_(n≥2) (1/(nln(n))) .

$${find}\:{the}\:{nature}\:{of}\:\:\:\sum_{{n}\geqslant\mathrm{2}} \:\frac{\mathrm{1}}{{nln}\left({n}\right)}\:. \\ $$

Question Number 31980 Answers: 0 Comments: 0

let −1<x<1 calculate Σ_(n=1) ^∞ (x^n /((1−x^n )(1−x^(n+1) ))) .

$${let}\:−\mathrm{1}<{x}<\mathrm{1}\:{calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{x}^{{n}} }{\left(\mathrm{1}−{x}^{{n}} \right)\left(\mathrm{1}−{x}^{{n}+\mathrm{1}} \right)}\:\:. \\ $$

Question Number 31979 Answers: 0 Comments: 1

calculate Σ_(n=2) ^∞ (1/((n^2 −1)^2 )) .

$${calculate}\:\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left({n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 31978 Answers: 1 Comments: 0

find the value of Σ_(n=0) ^∞ (1/((2n+1)(2n+3)(2n+5))).

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)\left(\mathrm{2}{n}+\mathrm{5}\right)}. \\ $$

Question Number 31977 Answers: 1 Comments: 2

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

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

Question Number 31976 Answers: 0 Comments: 0

let u_n =^(n+1) (√(n+1)) −^n (√n) find radius of convergence for Σ u_n z^n (z∈C).

$${let}\:{u}_{{n}} =^{{n}+\mathrm{1}} \sqrt{{n}+\mathrm{1}}\:−\:^{{n}} \sqrt{{n}}\:\:{find}\:{radius}\:{of}\:{convergence}\: \\ $$$${for}\:\:\Sigma\:{u}_{{n}} {z}^{{n}} \:\:\:\:\left({z}\in{C}\right). \\ $$

Question Number 31975 Answers: 0 Comments: 0

let u_n = ∫_1 ^∞ e^(−t^n ) dt 1) calculate lim_(n→∞) u_n 2)find a equivalent of u_n (n→∞) 3)find the radius of convergence of Σ u_n x^n .

$${let}\:{u}_{{n}} =\:\int_{\mathrm{1}} ^{\infty} \:\:{e}^{−{t}^{{n}} } \:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:\left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{radius}\:{of}\:{convergence}\:{of}\:\Sigma\:{u}_{{n}} {x}^{{n}} . \\ $$

Question Number 31974 Answers: 0 Comments: 0

1)find I(p,q) = ∫_0 ^1 t^p (1−t)^q dt with pand q integrs 2) find the nature of Σ I_((n,n))

$$\left.\mathrm{1}\right){find}\:{I}\left({p},{q}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{t}^{{p}} \:\left(\mathrm{1}−{t}\right)^{{q}} \:{dt}\:\:{with}\:{pand}\:{q}\:{integrs} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:\:{I}_{\left({n},{n}\right)} \\ $$

Question Number 31973 Answers: 0 Comments: 0

let give the sequence (u_n ) / u_0 =1 and u_1 =−1 and u_(n+2) = 2u_(n+1 ) −u_n .find the radius of convegence for this serie.

$${let}\:{give}\:{the}\:{sequence}\:\left({u}_{{n}} \right)\:\:/\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{\mathrm{1}} =−\mathrm{1}\:{and} \\ $$$${u}_{{n}+\mathrm{2}} =\:\mathrm{2}{u}_{{n}+\mathrm{1}\:} −{u}_{{n}} \:\:\:.{find}\:{the}\:{radius}\:{of}\:{convegence}\:{for} \\ $$$${this}\:{serie}. \\ $$

Question Number 31972 Answers: 0 Comments: 1

solve inside ]−1,1[ the d.e. (√(1−x^2 )) y^′ +y =e^(−2x) .

$$\left.{solve}\:{inside}\:\right]−\mathrm{1},\mathrm{1}\left[\:{the}\:{d}.{e}.\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:{y}^{'} \:+{y}\:={e}^{−\mathrm{2}{x}} \:.\right. \\ $$

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