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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) .

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

Question Number 31971 Answers: 0 Comments: 0

let consider the d.e. x(x−1)y^(′′) +3xy^′ +y =0 find a solution at form Σa_n x^n .

$${let}\:{consider}\:{the}\:{d}.{e}.\:{x}\left({x}−\mathrm{1}\right){y}^{''} \:+\mathrm{3}{xy}^{'} \:+{y}\:=\mathrm{0} \\ $$$${find}\:{a}\:{solution}\:{at}\:{form}\:\Sigma{a}_{{n}} {x}^{{n}} \:\:. \\ $$

Question Number 31970 Answers: 0 Comments: 0

find the nature of ∫_2 ^∞ (e^(−x) /(√(x^2 −4))) dx .

$${find}\:{the}\:{nature}\:{of}\:\:\int_{\mathrm{2}} ^{\infty} \:\:\frac{{e}^{−{x}} }{\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:{dx}\:. \\ $$

Question Number 31969 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (((1+t^2 )/(1+t^4 )))arctant dt.

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\left(\frac{\mathrm{1}+{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{4}} }\right){arctant}\:{dt}. \\ $$

Question Number 31968 Answers: 0 Comments: 0

find ∫_2 ^(√5) x(√((x−2)((√5)−x))) dx .

$${find}\:\:\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} {x}\sqrt{\left({x}−\mathrm{2}\right)\left(\sqrt{\mathrm{5}}−{x}\right)}\:{dx}\:. \\ $$

Question Number 31967 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((arctanx)/(x^2 +x+1))dx .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctanx}}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}{dx}\:. \\ $$

Question Number 31966 Answers: 0 Comments: 0

let give I_n = ∫_0 ^∞ (dt/((1+t^2 )^n )) with n integr and n≥1 1) prove the convergence of I_n 2)find lim_(n→∞) I_n 3) study the convergence of the serie Σ_(n=1) ^∞ (−1)^n I_n .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{{n}} }\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{convergence}\:{of}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow\infty} \:\:{I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:{the}\:{serie}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:\:{I}_{{n}} \:. \\ $$

Question Number 31965 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ (((−1)^(n−1) −2^n )/n) x^n with ∣x∣ <(1/2)

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:−\mathrm{2}^{{n}} }{{n}}\:{x}^{{n}} \:\:{with}\:\mid{x}\mid\:<\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 31964 Answers: 0 Comments: 0

1)find S_n = Σ_(k=0) ^n C_n ^k sin((k/n)) 2) study the convergence of S_n

$$\left.\mathrm{1}\right){find}\:\:{S}_{{n}} \:\:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:{sin}\left(\frac{{k}}{{n}}\right) \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:{S}_{{n}} \\ $$

Question Number 31963 Answers: 0 Comments: 0

find Re (((1+e^(iα) )/(1+e^(iβ) ))) and Im ( ((1+e^(iα) )/(1+e^(iβ) )) ) .

$${find}\:{Re}\:\left(\frac{\mathrm{1}+{e}^{{i}\alpha} }{\mathrm{1}+{e}^{{i}\beta} }\right)\:{and}\:{Im}\:\left(\:\frac{\mathrm{1}+{e}^{{i}\alpha} }{\mathrm{1}+{e}^{{i}\beta} }\:\right)\:. \\ $$

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