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Question Number 33593 Answers: 0 Comments: 0

calculate lim_(x→0) ((2(1−cosx)sinx −x^3 (1−x^2 )^(1/4) )/(sin^5 x −x^5 ))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\mathrm{2}\left(\mathrm{1}−{cosx}\right){sinx}\:−{x}^{\mathrm{3}} \:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{sin}^{\mathrm{5}} {x}\:−{x}^{\mathrm{5}} } \\ $$

Question Number 33592 Answers: 0 Comments: 0

let f(x) =e^(−x^2 ) 1) prove that f^((n)) (x) = p_n (x).e^(−x^2 ) where p_n is a polynome with deg=n 2) prove that ∀ n≥1 p_(n+1) (x) +α(x)p_n (x) +β(n)p_(n−1) (x) =0 find α and β 3)calculate p_0 ,p_1 ,p_2 ,p_3 4) calculate p_n ^(′′) (x) interms of p^′ (x) and p_n (x).

$${let}\:{f}\left({x}\right)\:={e}^{−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)\:=\:{p}_{{n}} \left({x}\right).{e}^{−{x}^{\mathrm{2}} } \:\:\:{where}\:{p}_{{n}} {is}\:{a}\:{polynome} \\ $$$${with}\:{deg}={n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\geqslant\mathrm{1}\: \\ $$$${p}_{{n}+\mathrm{1}} \left({x}\right)\:+\alpha\left({x}\right){p}_{{n}} \left({x}\right)\:+\beta\left({n}\right){p}_{{n}−\mathrm{1}} \left({x}\right)\:=\mathrm{0}\:\:{find}\:\alpha\:{and}\:\beta \\ $$$$\left.\mathrm{3}\right){calculate}\:{p}_{\mathrm{0}} ,{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{p}_{{n}} ^{''} \:\left({x}\right)\:{interms}\:{of}\:{p}^{'} \left({x}\right)\:{and}\:{p}_{{n}} \left({x}\right). \\ $$

Question Number 33591 Answers: 1 Comments: 1

find the value of Σ_(n=1) ^∞ (2/(n^3 +3n^2 +2n)) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{2}}{{n}^{\mathrm{3}} \:\:+\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{2}{n}}\:. \\ $$

Question Number 33590 Answers: 0 Comments: 1

let α >1 calculate f(α) = ∫_α ^(+∞) ((x^2 −x+1)/((x−1)^2 (x+1)^2 )) dx .

$${let}\:\alpha\:>\mathrm{1}\:\:{calculate}\:{f}\left(\alpha\right)\:=\:\int_{\alpha} ^{+\infty} \:\:\frac{{x}^{\mathrm{2}} −{x}+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 33589 Answers: 0 Comments: 1

1) decompose F(x) = (1/((x^2 +4)(x−3)^2 )) 2) calculate ∫_4 ^(+∞) (dx/((x^2 +4)(x−3)^2 )) .

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)\left({x}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{4}} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)\left({x}−\mathrm{3}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33588 Answers: 1 Comments: 1

find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)(2n+1))) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}\:. \\ $$

Question Number 33587 Answers: 0 Comments: 0

let f(x)=∫_0 ^π ln (x^2 −2x cosθ +1)dθ with ∣x∣<1 give a simple form of f(x).

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\:\left({x}^{\mathrm{2}} \:−\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${give}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 33586 Answers: 0 Comments: 0

find Σ_(n=1) ^∞ z^n ((sin(nθ))/n) with z from C and ∣z∣<1 .

$${find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{z}^{{n}} \:\:\frac{{sin}\left({n}\theta\right)}{{n}}\:\:{with}\:{z}\:{from}\:{C}\:{and}\:\mid{z}\mid<\mathrm{1}\:. \\ $$

Question Number 33585 Answers: 0 Comments: 0

study the nature of Σ_(n=1) ^∞ sin(πen!) .

$${study}\:{the}\:{nature}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} {sin}\left(\pi{en}!\right)\:. \\ $$

Question Number 33584 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ (((1+i)^n cos(nθ))/2^n ) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(\mathrm{1}+{i}\right)^{{n}} \:{cos}\left({n}\theta\right)}{\mathrm{2}^{{n}} }\:. \\ $$

Question Number 33583 Answers: 0 Comments: 0

let z∈C / ∣z∣<1 calculate Σ_(n=1) ^∞ z^n cos(nθ) and Σ_(n=1) ^∞ z^n sin(nθ)

$${let}\:{z}\in{C}\:\:/\:\mid{z}\mid<\mathrm{1}\:\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{z}^{{n}} {cos}\left({n}\theta\right) \\ $$$${and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{z}^{{n}} \:{sin}\left({n}\theta\right) \\ $$

Question Number 33582 Answers: 0 Comments: 0

find the nature of Σ_(n=1) ^∞ (n^(ln(n)) /e^(√n) ) .

$${find}\:{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}^{{ln}\left({n}\right)} }{{e}^{\sqrt{{n}}} }\:\:. \\ $$

Question Number 33581 Answers: 0 Comments: 0

find the nature of Σ_(n=1) ^∞ (−1)^n (−(1/n) +arctan((1/n)))^(1/3)

$${find}\:{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \left(−\frac{\mathrm{1}}{{n}}\:+{arctan}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$

Question Number 33575 Answers: 3 Comments: 0

simplify (√((3+2(√(2)))))

$$\mathrm{simplify}\:\sqrt{\left(\mathrm{3}+\mathrm{2}\sqrt{\left.\mathrm{2}\right)}\right.} \\ $$

Question Number 33571 Answers: 1 Comments: 0

f(x) = x^(20) + a_1 x^(19) + a_2 x^(18) + ... + a_(20) If f(1) = f(2) = f(3) = ... = f(20) What is the value of a_1 ?

$${f}\left({x}\right)\:=\:{x}^{\mathrm{20}} \:+\:{a}_{\mathrm{1}} {x}^{\mathrm{19}} \:+\:{a}_{\mathrm{2}} {x}^{\mathrm{18}} \:+\:...\:+\:{a}_{\mathrm{20}} \\ $$$$\mathrm{If}\:{f}\left(\mathrm{1}\right)\:=\:{f}\left(\mathrm{2}\right)\:=\:{f}\left(\mathrm{3}\right)\:=\:...\:=\:{f}\left(\mathrm{20}\right) \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{1}} \:? \\ $$

Question Number 33570 Answers: 1 Comments: 3

A = Σ_(n=2) ^(2017) [∫_1 ^n 2tan^(−1) x + sin^(−1) (((2x)/(1 + x^2 ))) dx] B = Π_(n=2) ^(2017) [∫_1 ^n 2tan^(−1) x + sin^(−1) (((2x)/(1 + x^2 ))) dx] A + B = ...

$${A}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\sum}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:{dx}\right] \\ $$$${B}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\prod}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:{dx}\right] \\ $$$${A}\:+\:{B}\:=\:... \\ $$

Question Number 33569 Answers: 1 Comments: 0

Given f(x) = x^3 + ax^2 + bx + c with a, b, c ∈ R, the roots are x_1 , x_2 , x_3 ∈ R Let λ is an positive integer that satisfied x_2 − x_1 = λ x_3 > (1/2)(x_1 + x_2 ) What is the max value of ((2a^3 + 27c − 9ab)/λ^3 ) ?

$$\mathrm{Given}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c} \\ $$$$\mathrm{with}\:{a},\:{b},\:{c}\:\in\:\mathbb{R},\:\mathrm{the}\:\mathrm{roots}\:\mathrm{are}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} \:\in\:\mathbb{R} \\ $$$$\mathrm{Let}\:\lambda\:\mathrm{is}\:\mathrm{an}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{satisfied} \\ $$$${x}_{\mathrm{2}} \:−\:{x}_{\mathrm{1}} \:=\:\lambda \\ $$$${x}_{\mathrm{3}} \:>\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}_{\mathrm{1}} \:+\:{x}_{\mathrm{2}} \right) \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{max}\:\mathrm{value}\:\mathrm{of}\:\:\frac{\mathrm{2}{a}^{\mathrm{3}} \:+\:\mathrm{27}{c}\:−\:\mathrm{9}{ab}}{\lambda^{\mathrm{3}} }\:? \\ $$

Question Number 33557 Answers: 0 Comments: 0

This^ is^ a^ formula. working^ out^ the^ area^ between^ two Toroidal^ coils. Both^ with^ a^ magnetic^ spring^ constant. r=radius=1 a=area=2 k^z =(1/((r+a)))−(1/r^2 )=−a(((2∙r+a))/([(r+a)^2 ∙r^2 ])) ratio=? what^ is^ the^ ratio?

$${This}^{} {is}^{} {a}^{} {formula}. \\ $$$${working}^{} {out}^{} {the}^{} {area}^{} {between}^{} {two} \\ $$$${Toroidal}^{} {coils}. \\ $$$${Both}^{} {with}^{} {a}^{} {magnetic}^{} {spring}^{} {constant}. \\ $$$$ \\ $$$${r}={radius}=\mathrm{1} \\ $$$$\mathrm{a}={area}=\mathrm{2} \\ $$$$ \\ $$$$\mathrm{k}^{\mathrm{z}} =\frac{\mathrm{1}}{\left(\mathrm{r}+\mathrm{a}\right)}−\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{2}} }=−\mathrm{a}\frac{\left(\mathrm{2}\centerdot\mathrm{r}+\mathrm{a}\right)}{\left[\left(\mathrm{r}+\mathrm{a}\right)^{\mathrm{2}} \centerdot\mathrm{r}^{\mathrm{2}} \right]} \\ $$$${ratio}=? \\ $$$${what}^{} {is}^{} {the}^{} {ratio}? \\ $$

Question Number 33551 Answers: 1 Comments: 1

Question Number 33544 Answers: 0 Comments: 0

1) find the value of ∫_0 ^∞ (( e^(−tx^2 ) )/(1+x^2 )) dx with t >0 2) find the value of ∫_0 ^∞ (((1−e^(−x^2 ) ))/(x^2 (1+x^2 )))dx .

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\:{e}^{−{tx}^{\mathrm{2}} } }{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{with}\:{t}\:>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}−{e}^{−{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\:. \\ $$

Question Number 33542 Answers: 0 Comments: 1

A car with a mass of 1300kg is constructed so that its frame is suspended by four strings.Each has a force constant of 20000N/m. Two people riding the car have a combined mass of 160kg.Find the frequency of vibration of the car.

$${A}\:{car}\:{with}\:{a}\:{mass}\:{of}\:\mathrm{1300}{kg}\:{is} \\ $$$${constructed}\:{so}\:{that}\:{its}\:{frame}\:{is}\: \\ $$$${suspended}\:{by}\:{four}\:{strings}.{Each} \\ $$$${has}\:{a}\:{force}\:{constant}\:{of}\:\mathrm{20000}{N}/{m}. \\ $$$${Two}\:{people}\:{riding}\:{the}\:{car}\:{have}\:{a} \\ $$$${combined}\:{mass}\:{of}\:\mathrm{160}{kg}.{Find}\:{the} \\ $$$${frequency}\:{of}\:{vibration}\:{of}\:{the}\:{car}. \\ $$

Question Number 33535 Answers: 1 Comments: 0

If the frequency of 0.75 long simple pendulum is 1.5Hz,the angular frequency on a corresponding reference circle in rad/s is a)1.5π b)3π c)0.5π d)2π

$${If}\:{the}\:{frequency}\:{of}\:\mathrm{0}.\mathrm{75}\:{long}\:{simple} \\ $$$${pendulum}\:{is}\:\mathrm{1}.\mathrm{5}{Hz},{the}\:{angular} \\ $$$${frequency}\:{on}\:{a}\:{corresponding} \\ $$$${reference}\:{circle}\:{in}\:{rad}/{s}\:{is} \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left..\mathrm{5}\pi\:{b}\right)\mathrm{3}\pi\:{c}\right)\mathrm{0}.\mathrm{5}\pi\:{d}\right)\mathrm{2}\pi \\ $$

Question Number 33533 Answers: 1 Comments: 0

A mass of 2kg is attached to a spring with constant 8N/m.It is⊛ then displaced to the point x=2. What time does it take for the block to travel to the point x=1? a)40s b)60s c)30s d)20s

$${A}\:{mass}\:{of}\:\mathrm{2}{kg}\:{is}\:{attached}\:{to}\:{a} \\ $$$${spring}\:{with}\:{constant}\:\mathrm{8}{N}/{m}.{It}\:{is}\circledast \\ $$$${then}\:{displaced}\:{to}\:{the}\:{point}\:{x}=\mathrm{2}. \\ $$$${What}\:{time}\:{does}\:{it}\:{take}\:{for}\:{the}\:{block} \\ $$$${to}\:{travel}\:{to}\:{the}\:{point}\:{x}=\mathrm{1}? \\ $$$$\left.{a}\left.\right)\left.\mathrm{4}\left.\mathrm{0}{s}\:{b}\right)\mathrm{60}{s}\:{c}\right)\mathrm{30}{s}\:{d}\right)\mathrm{20}{s} \\ $$

Question Number 33531 Answers: 0 Comments: 16

∫_0 ^∞ (e^(−x^2 ) /(x^2 +1))dx=?

$$\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}^{\mathrm{2}} } }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=? \\ $$

Question Number 33518 Answers: 0 Comments: 0

α^4 +β^(4 ) solve please

$$\alpha^{\mathrm{4}} +\beta^{\mathrm{4}\:} {solve}\:{please} \\ $$

Question Number 33515 Answers: 0 Comments: 1

expand α^4 +β^(β ) please

$${expand}\:\alpha^{\mathrm{4}} +\beta^{\beta\:\:} {please} \\ $$

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