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Question Number 32307 Answers: 0 Comments: 0

Question Number 32306 Answers: 0 Comments: 0

Question Number 32305 Answers: 1 Comments: 1

find ∫_1 ^e sin(ln(x))dx .

$${find}\:\int_{\mathrm{1}} ^{{e}} \:{sin}\left({ln}\left({x}\right)\right){dx}\:. \\ $$

Question Number 32304 Answers: 0 Comments: 0

find lim_(x→+∞) e^(−x^2 ) ∫_0 ^x e^t^2 dt .

$${find}\:{lim}_{{x}\rightarrow+\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{{t}^{\mathrm{2}} } {dt}\:\:. \\ $$

Question Number 32303 Answers: 0 Comments: 1

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

$${find}\:{lim}_{{n}\rightarrow\infty} \:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{2}{n}+{k}}\:\:. \\ $$

Question Number 32302 Answers: 1 Comments: 0

calculate ∫_1 ^2 (dx/(x +x(√x))) .

$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\:\frac{{dx}}{{x}\:+{x}\sqrt{{x}}}\:. \\ $$

Question Number 32301 Answers: 0 Comments: 1

calculate ∫_1 ^e ln(1+(√x))dx .

$${calculate}\:\int_{\mathrm{1}} ^{{e}} \:{ln}\left(\mathrm{1}+\sqrt{{x}}\right){dx}\:. \\ $$

Question Number 32300 Answers: 0 Comments: 0

calculate Σ_(n=0) ^∞ n^2 e^(inθ) 2) find Σ_(n=0) ^∞ n^2 cos(nθ) and Σ_(n=0) ^∞ n^2 sin(nθ)=.

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} {n}^{\mathrm{2}} \:{e}^{{in}\theta} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{n}^{\mathrm{2}} {cos}\left({n}\theta\right)\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{n}^{\mathrm{2}} {sin}\left({n}\theta\right)=. \\ $$

Question Number 32299 Answers: 0 Comments: 0

find tbe nature of the serie Σ_(n≥1) ((e^(1/n) +e^(−(1/n)) )/n) .

$${find}\:{tbe}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{{e}^{\frac{\mathrm{1}}{{n}}} \:\:+{e}^{−\frac{\mathrm{1}}{{n}}} }{{n}}\:. \\ $$$$ \\ $$

Question Number 32298 Answers: 0 Comments: 1

find tbe nature of Σ_(n≥2) (1/(n(ln(n))^2 )) .

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

Question Number 32297 Answers: 0 Comments: 0

calculate Σ_(n≥0) ((n+2^n )/(n!)) .

$${calculate}\:\sum_{{n}\geqslant\mathrm{0}} \:\frac{{n}+\mathrm{2}^{{n}} }{{n}!}\:\:. \\ $$

Question Number 32296 Answers: 0 Comments: 0

let u_n = (((n+1)^α −n^α )/n^(α−1) ) with α>1 find lim_(n→∞) u_n .

$${let}\:\:{u}_{{n}} =\:\frac{\left({n}+\mathrm{1}\right)^{\alpha} \:\:−{n}^{\alpha} }{{n}^{\alpha−\mathrm{1}} }\:\:{with}\:\alpha>\mathrm{1}\:\:{find}\:{lim}_{{n}\rightarrow\infty} {u}_{{n}} \:\:. \\ $$

Question Number 32295 Answers: 0 Comments: 2

calculate Σ_(k=0) ^n (2k+1)(−1)^k .

$${calculate}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(\mathrm{2}{k}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{k}} \:\:. \\ $$

Question Number 32294 Answers: 0 Comments: 1

let u_1 =1 and u_2 =2 and u_n =u_(n−1) +u_(n−2) find u_n interms of n .

$${let}\:{u}_{\mathrm{1}} =\mathrm{1}\:{and}\:{u}_{\mathrm{2}} =\mathrm{2}\:{and}\:{u}_{{n}} ={u}_{{n}−\mathrm{1}} \:+{u}_{{n}−\mathrm{2}} \\ $$$${find}\:{u}_{{n}} \:{interms}\:{of}\:{n}\:. \\ $$

Question Number 32293 Answers: 1 Comments: 1

let u_0 = (√3) and u_(n+1) =(√(2+u_n ^2 )) calculate u_n interms of n.

$${let}\:{u}_{\mathrm{0}} =\:\sqrt{\mathrm{3}}\:\:{and}\:{u}_{{n}+\mathrm{1}} =\sqrt{\mathrm{2}+{u}_{{n}} ^{\mathrm{2}} } \\ $$$${calculate}\:{u}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 32291 Answers: 0 Comments: 0

let u_n = Σ_(k=1) ^n (1/(n+k)) prove that 0≤u_n ≤1 .

$${let}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{n}+{k}}\:{prove}\:{that}\:\mathrm{0}\leqslant{u}_{{n}} \leqslant\mathrm{1}\:. \\ $$

Question Number 32290 Answers: 0 Comments: 1

let give u_0 =1 and u_(n+1) =(√(1+(√u_n ))) prove that u_n is increasing .

$${let}\:{give}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{{n}+\mathrm{1}} =\sqrt{\mathrm{1}+\sqrt{{u}_{{n}} }}\:\:{prove}\:{that}\:{u}_{{n}} \:{is} \\ $$$${increasing}\:. \\ $$

Question Number 32289 Answers: 0 Comments: 0

find lim_(x→0) ln ( ((e^(2x) −1)/x)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:{ln}\:\left(\:\frac{{e}^{\mathrm{2}{x}} −\mathrm{1}}{{x}}\right)\:. \\ $$

Question Number 32288 Answers: 0 Comments: 0

study the function f(x)=(x^2 /(x+1)) e^(1/x) .

$${study}\:{the}\:{function}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}+\mathrm{1}}\:{e}^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 32287 Answers: 0 Comments: 0

1) for x>0 prove that (1/(x+1)) ≤ln(x+1)−lnx ≤ (1/x) 2) let u_n = Σ_(p=1) ^(kn) (1/p) find lim_(n→∞ ) u_n .

$$\left.\mathrm{1}\right)\:{for}\:{x}>\mathrm{0}\:{prove}\:{that}\:\frac{\mathrm{1}}{{x}+\mathrm{1}}\:\leqslant{ln}\left({x}+\mathrm{1}\right)−{lnx}\:\leqslant\:\frac{\mathrm{1}}{{x}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\sum_{{p}=\mathrm{1}} ^{{kn}} \:\frac{\mathrm{1}}{{p}}\:\:\:{find}\:{lim}_{{n}\rightarrow\infty\:} \:{u}_{{n}} . \\ $$

Question Number 32286 Answers: 0 Comments: 0

calculate lim_(x→0) (((e^x /(√(1+x))) −1−(x/2))/x^2 ) .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\frac{{e}^{{x}} }{\sqrt{\mathrm{1}+{x}}}\:−\mathrm{1}−\frac{{x}}{\mathrm{2}}}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32281 Answers: 0 Comments: 0

calculate lim_(x→∞) (√(x^2 +x+1)) −(√(x^2 −x+1)) .

$${calculate}\:{lim}_{{x}\rightarrow\infty} \sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:−\sqrt{{x}^{\mathrm{2}} \:−{x}+\mathrm{1}}\:\:. \\ $$

Question Number 32280 Answers: 0 Comments: 0

let u_n = e^(1/(n^2 +1)) −1 1) find a equivalent of u_n and lim_(n→∞) u_n 2) study the convergence of Σu_n .

$${let}\:{u}_{{n}} =\:{e}^{\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\mathrm{1}}} \:\:−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:{and}\:{lim}_{{n}\rightarrow\infty} {u}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\:\Sigma{u}_{{n}} \:. \\ $$

Question Number 32279 Answers: 0 Comments: 0

find lim_(x→0) ((tan^2 x)/((1−cosx))) .((e^x −1)/x)

$${find}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{tan}^{\mathrm{2}} {x}}{\left(\mathrm{1}−{cosx}\right)}\:.\frac{{e}^{{x}} \:−\mathrm{1}}{{x}}\:\:\: \\ $$

Question Number 32282 Answers: 0 Comments: 0

let give f(x)=(√(1+x^2 )) find f^((n)) (o) .

$${let}\:{give}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\:\:{find}\:\:{f}^{\left({n}\right)} \left({o}\right)\:.\: \\ $$

Question Number 32278 Answers: 0 Comments: 0

calculate lim_(x→+∞) (x−1)cos((π/x)) .

$${calculate}\:{lim}_{{x}\rightarrow+\infty} \left({x}−\mathrm{1}\right){cos}\left(\frac{\pi}{{x}}\right)\:. \\ $$

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