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Question Number 34692 Answers: 0 Comments: 1

find lim_(n→+∞) (1/n^3 ) Σ_(k=1) ^n k^2 sin(((kπ)/n))

$${find}\:{lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}^{\mathrm{3}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{2}} \:{sin}\left(\frac{{k}\pi}{{n}}\right) \\ $$

Question Number 34691 Answers: 0 Comments: 0

calculate Σ_(n=3) ^∞ ((2n−1)/(n^3 −4n)) .

$${calculate}\:\:\sum_{{n}=\mathrm{3}} ^{\infty} \:\:\:\frac{\mathrm{2}{n}−\mathrm{1}}{{n}^{\mathrm{3}} \:−\mathrm{4}{n}}\:. \\ $$

Question Number 34690 Answers: 0 Comments: 0

let U_n =(1/(n!)) ∫_0 ^1 (arcsinx)^n dx calculate lim_(n→+∞) U_n .

$${let}\:{U}_{{n}} =\frac{\mathrm{1}}{{n}!}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left({arcsinx}\right)^{{n}} {dx} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:. \\ $$

Question Number 34689 Answers: 0 Comments: 1

prove that Σ_(k=1) ^n sin((k/n^2 )) =(1/2) +(1/(2n)) +o((1/n))

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}{n}}\:+{o}\left(\frac{\mathrm{1}}{{n}}\right) \\ $$

Question Number 34688 Answers: 0 Comments: 1

cslculate Σ_(n=2) ^∞ ln(1+(((−1)^n )/n))

$${cslculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\right) \\ $$

Question Number 34687 Answers: 0 Comments: 0

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

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

Question Number 34686 Answers: 0 Comments: 0

decompose F(x) = (((2n)!)/((x^2 −1)(x^2 −2)....(x^2 −n)))

$${decompose}\:{F}\left({x}\right)\:\:=\:\frac{\left(\mathrm{2}{n}\right)!}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} \:−\mathrm{2}\right)....\left({x}^{\mathrm{2}} \:−{n}\right)} \\ $$

Question Number 34685 Answers: 0 Comments: 0

decompose the fraction F(x)= (1/((x+2)( x^n −1))) with n ∈ N^★

$${decompose}\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}+\mathrm{2}\right)\left(\:{x}^{{n}} \:\:−\mathrm{1}\right)}\:\:{with}\:{n}\:\in\:{N}^{\bigstar} \\ $$

Question Number 34684 Answers: 0 Comments: 0

let U_n = (π/4) −Σ_(k=0) ^n (((−1)^k )/(2k+1)) calcilate Σ_(n=0) ^∞ U_n

$${let}\:{U}_{{n}} =\:\frac{\pi}{\mathrm{4}}\:−\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$${calcilate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 34683 Answers: 0 Comments: 0

find?the nature of Σ_(n=0) ^∞ sin{π(2+(√3) )^n }

$${find}?{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left\{\pi\left(\mathrm{2}+\sqrt{\mathrm{3}}\:\right)^{{n}} \right\} \\ $$

Question Number 34682 Answers: 0 Comments: 0

calculate Σ_(n=0) ^∞ ln(cos((a/2^n )))

$${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:{ln}\left({cos}\left(\frac{{a}}{\mathrm{2}^{{n}} }\right)\right) \\ $$

Question Number 34681 Answers: 0 Comments: 0

calculate lim_(x→0) { ((1+tanx)/(1+thx))}^(1/(sinx)) .

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\left\{\:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{thx}}\right\}^{\frac{\mathrm{1}}{{sinx}}} . \\ $$

Question Number 34680 Answers: 0 Comments: 0

decompose inside C(x) the fraction F(x) = (x^2 /(x^4 −2x^2 cos(2a) +1)) .

$${decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)\:=\:\:\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} {cos}\left(\mathrm{2}{a}\right)\:+\mathrm{1}}\:. \\ $$

Question Number 34679 Answers: 0 Comments: 0

let f(x) = (x/(4x^2 −1)) 1) find f^((n)) (x) and f^((n)) (0) 2) developp f at ontegr serie .

$${let}\:{f}\left({x}\right)\:=\:\:\frac{{x}}{\mathrm{4}{x}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{ontegr}\:{serie}\:. \\ $$

Question Number 34678 Answers: 0 Comments: 0

prove that ∀ n≥3 (√n) <^n (√(n!))

$${prove}\:{that}\:\forall\:{n}\geqslant\mathrm{3}\:\:\:\:\:\sqrt{{n}}\:\:<^{{n}} \sqrt{{n}!} \\ $$$$ \\ $$

Question Number 34677 Answers: 0 Comments: 0

prove that Σ_(k=0) ^(n−1) [x +(k/n)] =[nx] ∀ n∈ ∈N^★

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\left[{x}\:+\frac{{k}}{{n}}\right]\:=\left[{nx}\right]\:\:\forall\:{n}\in\:\in{N}^{\bigstar} \\ $$

Question Number 34676 Answers: 0 Comments: 0

prove that Σ_(k=0) ^(2n−1) (((−1)^k )/(k+1)) =Σ_(k=n+1) ^(2n) (1/k)

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

Question Number 34675 Answers: 0 Comments: 0

provethat e = Σ_(k=0) ^n (1/(k!)) +∫_0 ^1 (((1−t)^n )/(n!)) e^t dt .

$${provethat}\:{e}\:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}}{{k}!}\:\:+\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\left(\mathrm{1}−{t}\right)^{{n}} }{{n}!}\:{e}^{{t}} \:{dt}\:. \\ $$

Question Number 34674 Answers: 0 Comments: 0

find ∫_0 ^π ((x sinx)/(1+cos^2 x)) dx

$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}\:{sinx}}{\mathrm{1}+{cos}^{\mathrm{2}} {x}}\:{dx} \\ $$

Question Number 34673 Answers: 0 Comments: 0

solve (((1+iz)/(1−iz)))^n = ((1+itanα)/(1−itanα)) with −(π/2)<α<(π/2)

$${solve}\:\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right)^{{n}} \:=\:\frac{\mathrm{1}+{itan}\alpha}{\mathrm{1}−{itan}\alpha}\:\:{with}\:−\frac{\pi}{\mathrm{2}}<\alpha<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 34672 Answers: 0 Comments: 0

prove that ∀n∈N ∣sin(nx)∣≤n∣sinx∣ .

$${prove}\:{that}\:\forall{n}\in{N}\:\:\:\mid{sin}\left({nx}\right)\mid\leqslant{n}\mid{sinx}\mid\:. \\ $$

Question Number 34671 Answers: 0 Comments: 0

calculste Σ_(n=1) ^∞ arctan((2/n^2 )).

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

Question Number 34669 Answers: 0 Comments: 1

let P(x)=(1+x+ix^2 )^n −(1+x −ix^2 )^n 1) find the roots of P(x) 2) factorize inside C[x] P(x) 3) factorize indide R[x] P(x).

$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{x}+{ix}^{\mathrm{2}} \right)^{{n}} \:−\left(\mathrm{1}+{x}\:−{ix}^{\mathrm{2}} \right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{P}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{indide}\:{R}\left[{x}\right]\:{P}\left({x}\right). \\ $$

Question Number 34668 Answers: 0 Comments: 0

find the roots of?p(x) = x^(2n) −2x^n cos(nθ) +1 2)?factorize p(x)

$${find}\:{the}\:{roots}\:{of}?{p}\left({x}\right)\:=\:{x}^{\mathrm{2}{n}} \:−\mathrm{2}{x}^{{n}} \:{cos}\left({n}\theta\right)\:+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)?{factorize}\:{p}\left({x}\right)\: \\ $$

Question Number 34667 Answers: 0 Comments: 0

solve (x+1)^n = e^(2ina) then find the value of P_n = Π_(k=0) ^(n−1) sin(a +((kπ)/n))

$${solve}\:\:\left({x}+\mathrm{1}\right)^{{n}} \:=\:{e}^{\mathrm{2}{ina}} \:\:\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$${P}_{{n}} =\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{sin}\left({a}\:+\frac{{k}\pi}{{n}}\right) \\ $$

Question Number 34666 Answers: 0 Comments: 0

simplify sin^2 ( ((arccosx)/2))

$${simplify}\:{sin}^{\mathrm{2}} \left(\:\frac{{arccosx}}{\mathrm{2}}\right) \\ $$

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