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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) .

$$\left.{let}\:{give}\:\theta\in\right]\mathrm{0},\pi\left[\:\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{e}^{−{i}\theta} −{t}}=\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{{in}\theta} }{{n}}\:\:.\right. \\ $$

Question Number 28743    Answers: 1   Comments: 2

find the next 4 term and the n^(th) term 1,2,5,26.......

$${find}\:{the}\:{next}\:\mathrm{4}\:{term}\:{and}\:{the} \\ $$$${n}^{{th}} \:{term} \\ $$$$\mathrm{1},\mathrm{2},\mathrm{5},\mathrm{26}....... \\ $$

Question Number 28600    Answers: 0   Comments: 4

Question Number 28597    Answers: 1   Comments: 1

Question Number 28591    Answers: 1   Comments: 0

What is the difference between angular frequency and angular velocity?

$${What}\:{is}\:{the}\:{difference}\:{between} \\ $$$${angular}\:{frequency}\:{and}\:{angular} \\ $$$${velocity}? \\ $$

Question Number 28583    Answers: 1   Comments: 1

Question Number 28574    Answers: 1   Comments: 2

Question Number 28586    Answers: 0   Comments: 0

When dry chlorine is passed thru silver chlorate heated to 90°C then which of the oxide of chlorine is formed and why?

$$\mathrm{When}\:\mathrm{dry}\:\mathrm{chlorine}\:\mathrm{is}\:\mathrm{passed}\:\mathrm{thru} \\ $$$$\mathrm{silver}\:\mathrm{chlorate}\:\mathrm{heated}\:\mathrm{to}\:\mathrm{90}°\mathrm{C} \\ $$$$\mathrm{then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{oxide}\:\mathrm{of}\:\mathrm{chlorine} \\ $$$$\mathrm{is}\:\mathrm{formed}\:\mathrm{and}\:\mathrm{why}? \\ $$

Question Number 28585    Answers: 1   Comments: 0

Question Number 28565    Answers: 1   Comments: 0

Question Number 28554    Answers: 2   Comments: 0

(((a − b))/((c − d))) = 3 (((a − c))/((b − d))) = 4 (((a − d))/((b − c))) = ?

$$\frac{\left({a}\:−\:{b}\right)}{\left({c}\:−\:{d}\right)}\:\:=\:\:\mathrm{3} \\ $$$$\frac{\left({a}\:−\:{c}\right)}{\left({b}\:−\:{d}\right)}\:\:=\:\:\mathrm{4} \\ $$$$\frac{\left({a}\:−\:{d}\right)}{\left({b}\:−\:{c}\right)}\:\:=\:\:? \\ $$$$ \\ $$

Question Number 28547    Answers: 1   Comments: 0

Question Number 28546    Answers: 0   Comments: 0

let give w=e^(i((2π)/n)) and S= Σ_(k=0) ^(n−1) w^k^2 1) prove that S= Σ_(k=0) ^(n−1) w^((q+k)^2 ) 2) find ∣S∣.

$${let}\:{give}\:{w}={e}^{{i}\frac{\mathrm{2}\pi}{{n}}} \:\:\:{and}\:\:{S}=\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{w}^{{k}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:{S}=\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{w}^{\left({q}+{k}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:\mid{S}\mid. \\ $$

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