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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

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

Question Number 33513 Answers: 0 Comments: 1

In a recent pool of 500 men and 500 women it was observed that a total of 650 were married. of those married 275 were men and 500 claimed to be happy.out of 750 who claimed to be happy,400 were men of which 200 were married.Represent this information on the venn−diagram and then find (i)the number of married who are unhappy. (ii)the number of married who are happy.

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