Question and Answers Forum

AllQuestion and Answers: Page 1789

Question Number 28620 Answers: 0 Comments: 4

calculate Σ_(n=p) ^(+∞) C_(n ) ^p x^n .

$${calculate}\:\:\sum_{{n}={p}} ^{+\infty} \:\:\:{C}_{{n}\:} ^{{p}} {x}^{{n}} . \\ $$

Question Number 28619 Answers: 0 Comments: 1

calculate Σ_(k=2) ^(+∞) ln(1−(1/k^2 )) .

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

Question Number 28618 Answers: 0 Comments: 0

let give u_n = Σ_(k=n) ^(+∞) (((−1)^k )/(√(k+1))) study the convergence of Σ u_n .

$${let}\:{give}\:{u}_{{n}} =\:\sum_{{k}={n}} ^{+\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{{k}+\mathrm{1}}}\:\:{study}\:{the}\:{convergence}\:{of}\: \\ $$$$\Sigma\:{u}_{{n}} . \\ $$

Question Number 28617 Answers: 0 Comments: 0

let give a sequence of reals (a_n )_n / a_n >0 and U_n = (a_n /((1+a_1 )(1+a_2 )....(1+a_n ))) 1) prove that Σ u_n converges 2) calculate Σ u_n if u_n = (1/(√n)) .

$${let}\:{give}\:{a}\:{sequence}\:{of}\:{reals}\:\left({a}_{{n}} \right)_{{n}} \:\:/\:{a}_{{n}} >\mathrm{0}\:\:{and} \\ $$$${U}_{{n}} =\:\:\:\frac{{a}_{{n}} }{\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)....\left(\mathrm{1}+{a}_{{n}} \right)} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Sigma\:{u}_{{n}} \:{converges} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Sigma\:{u}_{{n}} \:\:{if}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\sqrt{{n}}}\:. \\ $$

Question Number 28616 Answers: 0 Comments: 0

let give u_n = (1+(1/n))^n −e find nature of Σ u_n .

$${let}\:{give}\:{u}_{{n}} =\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \:−{e}\:\:\:{find}\:{nature}\:{of}\:\:\Sigma\:{u}_{{n}} . \\ $$

Question Number 28615 Answers: 0 Comments: 0

find ∫_0 ^∞ ((shx)/x) e^(−3x) dx .

$${find}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{shx}}{{x}}\:{e}^{−\mathrm{3}{x}} {dx}\:. \\ $$

Question Number 28614 Answers: 0 Comments: 2

find Σ_(n=1) ^∞ ((sin(nx))/n).

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

Question Number 28613 Answers: 0 Comments: 1

let give x>0 and S(x)= ∫_0 ^∞ ((sint)/(e^(xt) −1))dt . developp S at form of series.

$${let}\:{give}\:{x}>\mathrm{0}\:\:{and}\:{S}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sint}}{{e}^{{xt}} −\mathrm{1}}{dt}\:. \\ $$$${developp}\:{S}\:{at}\:{form}\:{of}\:{series}. \\ $$

Question Number 28612 Answers: 0 Comments: 0

let give u_(n ) = Σ_(k=1) ^n ((sin(kα))/(n+k)) and α∈R find lim _(n→+∞) u_n .

$${let}\:{give}\:\:{u}_{{n}\:} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{sin}\left({k}\alpha\right)}{{n}+{k}}\:{and}\:\:\alpha\in{R} \\ $$$${find}\:{lim}\:_{{n}\rightarrow+\infty} {u}_{{n}} \:\:. \\ $$

Question Number 28610 Answers: 0 Comments: 0

let give I(x)= ∫_0 ^(π/2) (dt/(√(sin^2 t +x^2 cos^2 t))) and J(x)= ∫_0 ^(π/2) ((cost)/(√(sin^2 t +x^2 cos^2 t)))dt cslculate lim_(x→0^+ ) (I(x)−J(x)) and prove that I(x)=_(x→0^+ ) −lnx +2ln2 +o(1).

$${let}\:{give}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} \:{cos}^{\mathrm{2}} {t}}}\:\:{and} \\ $$$${J}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cost}}{\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} {cos}^{\mathrm{2}} {t}}}{dt}\:{cslculate}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \left({I}\left({x}\right)−{J}\left({x}\right)\right) \\ $$$${and}\:{prove}\:{that}\:\:{I}\left({x}\right)=_{{x}\rightarrow\mathrm{0}^{+} } \:−{lnx}\:+\mathrm{2}{ln}\mathrm{2}\:+{o}\left(\mathrm{1}\right). \\ $$

Question Number 28609 Answers: 0 Comments: 0

calculate cotanx −2cotan(2x)then simlify Σ_(k=0) ^n (1/2^k )tan((α/2^k )).

$${calculate}\:{cotanx}\:−\mathrm{2}{cotan}\left(\mathrm{2}{x}\right){then}\:{simlify} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{2}^{{k}} }{tan}\left(\frac{\alpha}{\mathrm{2}^{{k}} }\right). \\ $$$$ \\ $$

Question Number 28608 Answers: 1 Comments: 0

transform tanp−tanq then find the value of Σ_(k=1) ^n (1/(cos(kθ)cos((k+1)θ)) . θ∈R.

$${transform}\:{tanp}−{tanq}\:{then}\:{find}\:{the}\:{value}\:{of}\: \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{cos}\left({k}\theta\right){cos}\left(\left({k}+\mathrm{1}\right)\theta\right.}\:.\:\:\theta\in{R}. \\ $$

Question Number 28607 Answers: 0 Comments: 0

simplify Σ_(k=0) ^(n−1) 3^k sin^3 ((α/3^(k+1) )) .

$${simplify}\:\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\mathrm{3}^{{k}} \:{sin}^{\mathrm{3}} \left(\frac{\alpha}{\mathrm{3}^{{k}+\mathrm{1}} }\right)\:. \\ $$$$ \\ $$

Question Number 28611 Answers: 0 Comments: 1

let give θ∈]0,π[ prove that ∫_0 ^1 (dt/(e^(−iθ) −t))= Σ_(n=1) ^(+∞) (e^(inθ) /n) .