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AllQuestion and Answers: Page 1699

Question Number 30553 Answers: 0 Comments: 3

find lim_(x→0) (1+sinx)^x −(1+x)^(sinx) .

$${find}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \left(\mathrm{1}+{sinx}\right)^{{x}} \:−\left(\mathrm{1}+{x}\right)^{{sinx}} . \\ $$

Question Number 30552 Answers: 0 Comments: 0

find s(x)= Σ_(n≥0) ((sin(na))/((sina)^n )) (x^n /(n!)) and T(x) =Σ_(n≥0) ((cos(na))/((sina)^n )) (x^n /(n!)) .

$${find}\:{s}\left({x}\right)=\:\sum_{{n}\geqslant\mathrm{0}} \:\frac{{sin}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:{and}\: \\ $$$${T}\left({x}\right)\:=\sum_{{n}\geqslant\mathrm{0}} \:\:\frac{{cos}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:. \\ $$

Question Number 30551 Answers: 0 Comments: 0

find S = Σ_(n≥3) (1/((n+1)(n−2)2^n )) .

$$\:{find}\:\:{S}\:=\:\sum_{{n}\geqslant\mathrm{3}} \:\:\:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}−\mathrm{2}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 30550 Answers: 0 Comments: 0

let f(z)= Σ_(n≥0) a_n z^n /a_0 =1 ,a_1 =3 and ∀n≥2 a_n =3a_(n−1) −2 a_(n−2) find f(z) for ∣z∣<1 (z∈C) .

$${let}\:{f}\left({z}\right)=\:\sum_{{n}\geqslant\mathrm{0}} {a}_{{n}} {z}^{{n}} \:\:\:/{a}_{\mathrm{0}} =\mathrm{1}\:,{a}_{\mathrm{1}} =\mathrm{3}\:{and}\:\forall{n}\geqslant\mathrm{2} \\ $$$${a}_{{n}} =\mathrm{3}{a}_{{n}−\mathrm{1}} −\mathrm{2}\:{a}_{{n}−\mathrm{2}} \:\:\:\:{find}\:{f}\left({z}\right)\:{for}\:\mid{z}\mid<\mathrm{1}\:\:\left({z}\in{C}\right)\:. \\ $$

Question Number 30549 Answers: 0 Comments: 0

let S(x)= Σ_(n=0) ^∞ (x^(3n) /((3n)!)) find S(x).

$${let}\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}{n}} }{\left(\mathrm{3}{n}\right)!}\:\:{find}\:{S}\left({x}\right). \\ $$

Question Number 30548 Answers: 0 Comments: 0

let put for ∣λ∣<1 u_n = ∫_0 ^π ((cos(nx))/(1−2λcosx +λ^2 ))dx find u_n interms of n and λ.

$${let}\:{put}\:\:{for}\:\mid\lambda\mid<\mathrm{1}\:\:\:\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{1}−\mathrm{2}\lambda{cosx}\:+\lambda^{\mathrm{2}} }{dx}\: \\ $$$${find}\:{u}_{{n}} \:{interms}\:{of}\:{n}\:{and}\:\lambda. \\ $$

Question Number 30547 Answers: 0 Comments: 0

ind S= Σ_(n=0) ^∞ ((n^3 +n^2 +n+1)/(n!)) .

$${ind}\:{S}=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{n}^{\mathrm{3}} \:+{n}^{\mathrm{2}} \:+{n}+\mathrm{1}}{{n}!}\:\:. \\ $$

Question Number 30546 Answers: 0 Comments: 0

find I = ∫_1 ^(+∞) (((−1)^([x]) )/x^2 )dx .

$${find}\:{I}\:=\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\left(−\mathrm{1}\right)^{\left[{x}\right]} }{{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 30545 Answers: 0 Comments: 0

Question Number 30544 Answers: 0 Comments: 0

find I= ∫_0 ^π (t/(2+sint))dt.

$${find}\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{t}}{\mathrm{2}+{sint}}{dt}. \\ $$

Question Number 30543 Answers: 1 Comments: 1

Question Number 30542 Answers: 0 Comments: 0

prove that ∫_0 ^x e^(−u^2 ) du= x ∫_0 ^(π/4) (e^(−x^2 tan^2 t) /(cos^2 t))dt .

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{−{u}^{\mathrm{2}} } {du}=\:{x}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{e}^{−{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {t}} }{{cos}^{\mathrm{2}} {t}}{dt}\:\:. \\ $$$$ \\ $$

Question Number 30541 Answers: 0 Comments: 0

4n568

$$\mathrm{4}{n}\mathrm{568} \\ $$

Question Number 30540 Answers: 0 Comments: 0

Show that determinant (((bc ca ab)),(( a b c)),(( a^(2 ) b^2 c^2 ))) = (b − a)(c − a)(c − b)(ab + bc + ac)

$$\mathrm{Show}\:\mathrm{that}\:\:\begin{vmatrix}{\mathrm{bc}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{ca}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{ab}}\\{\:\mathrm{a}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{b}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{c}}\\{\:\mathrm{a}^{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} \:\mathrm{b}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{c}^{\mathrm{2}} }\end{vmatrix}\:\:=\:\:\left(\mathrm{b}\:−\:\mathrm{a}\right)\left(\mathrm{c}\:−\:\mathrm{a}\right)\left(\mathrm{c}\:−\:\mathrm{b}\right)\left(\mathrm{ab}\:+\:\mathrm{bc}\:+\:\mathrm{ac}\right) \\ $$

Question Number 30538 Answers: 1 Comments: 0

A man rows a boat downstream for 3 hours and then upstream for 3 hours. If he covered a total distance of 12km, find the speed of the water current.

$${A}\:{man}\:{rows}\:{a}\:{boat}\:{downstream} \\ $$$${for}\:\mathrm{3}\:{hours}\:{and}\:{then}\:{upstream} \\ $$$${for}\:\mathrm{3}\:{hours}.\:{If}\:{he}\:{covered}\:{a} \\ $$$${total}\:{distance}\:{of}\:\mathrm{12}{km},\:{find} \\ $$$${the}\:{speed}\:{of}\:{the}\:{water}\:{current}. \\ $$$$ \\ $$

Question Number 30626 Answers: 0 Comments: 6

Question Number 30529 Answers: 0 Comments: 0

let f(x)=e^(−x^2 ) prove that f^((n)) is at form f^((n)) = p_(n ) e^(−x^2 ) find relation between p_n and p_(n+1 ) . 2) find p_0 ,p_1 , p_2 ,p_3

$${let}\:{f}\left({x}\right)={e}^{−{x}^{\mathrm{2}} } \:\:{prove}\:{that}\:{f}^{\left({n}\right)} \:{is}\:{at}\:{form} \\ $$$${f}^{\left({n}\right)} =\:{p}_{{n}\:} \:{e}^{−{x}^{\mathrm{2}} } \:\:{find}\:{relation}\:{between}\:{p}_{{n}} {and}\:{p}_{{n}+\mathrm{1}\:} . \\ $$$$\left.\mathrm{2}\right)\:{find}\:{p}_{\mathrm{0}} \:,{p}_{\mathrm{1}} ,\:{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} \\ $$

Question Number 30528 Answers: 1 Comments: 1

simplify A= arctan(((sinx)/(1−cosx))) .

$${simplify}\:\:{A}=\:{arctan}\left(\frac{{sinx}}{\mathrm{1}−{cosx}}\right)\:. \\ $$

Question Number 30527 Answers: 1 Comments: 0

find I_(n,p) = ∫_0 ^∞ x^n e^(−px) with n and p from N^★ .

$${find}\:\:{I}_{{n},{p}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{{n}} \:{e}^{−{px}} \:\:\:\:\:{with}\:{n}\:{and}\:{p}\:{from}\:{N}^{\bigstar} \:. \\ $$

Question Number 30526 Answers: 0 Comments: 0

let f(x)=(√(1+ax)) with a∈C find f^((n)) (x) and f^((n)) (0) 2) developp f(x) at integr series.

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{ax}}\:\:{with}\:{a}\in{C}\:\:\:{find}\:\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\left({x}\right)\:{at}\:{integr}\:{series}. \\ $$

Question Number 30525 Answers: 0 Comments: 0

let I = ∫_0 ^∞ (e^(−x) /(1+x^2 )) give I at form of series .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{give}\:{I}\:{at}\:{form}\:{of}\:{series}\:. \\ $$

Question Number 30524 Answers: 1 Comments: 2

let w_n = Σ_(k=2) ^n (1/(k^2 −1)) find lim_(n→∞) w_n .

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

Question Number 30523 Answers: 0 Comments: 0

(α_k )_(0≤k≤n−1) are roots of x^n −1 simplify Π_n =Π_(k=0) ^(n−1) (x+α_k y) .

$$\:\left(\alpha_{{k}} \right)_{\mathrm{0}\leqslant{k}\leqslant{n}−\mathrm{1}} {are}\:{roots}\:{of}\:\:{x}^{{n}} −\mathrm{1}\:\:{simplify} \\ $$$$\prod_{{n}} =\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\left({x}+\alpha_{{k}} {y}\right)\:. \\ $$

Question Number 30522 Answers: 1 Comments: 1

let p(x)= (1+ix)^n −(1−ix)^n 1) find the roots of p(x) and factorize p(x) ) give p(x) at form of arcs.

$${let}\:{p}\left({x}\right)=\:\left(\mathrm{1}+{ix}\right)^{{n}} \:−\left(\mathrm{1}−{ix}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:{and}\:{factorize}\:{p}\left({x}\right) \\ $$$$\left.\right)\:{give}\:{p}\left({x}\right)\:{at}\:{form}\:{of}\:{arcs}. \\ $$$$ \\ $$

Question Number 30521 Answers: 0 Comments: 2

1) find ∫_0 ^1 ((√(1+x^2 )))^n cos(narctanx)dx 2)find ∫_0 ^1 ((√(1+x^2 )))^3 cos(3 arctanx)dx .

$$\left.\mathrm{1}\right)\:{find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}} \:{cos}\left({narctanx}\right){dx} \\ $$$$\left.\mathrm{2}\right){find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\right)^{\mathrm{3}} \:{cos}\left(\mathrm{3}\:{arctanx}\right){dx}\:. \\ $$

Question Number 30519 Answers: 1 Comments: 0

let j=e^(i((2π)/3)) find the value of Σ_(k=0) ^n C_n ^k (1+j)^k j^(2n−2k) .

$${let}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\:\:{find}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} {C}_{{n}} ^{{k}} \left(\mathrm{1}+{j}\right)^{{k}} {j}^{\mathrm{2}{n}−\mathrm{2}{k}} \:. \\ $$

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