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Question Number 26111 Answers: 0 Comments: 2

find the radius of convergence for the serie Σ_(n=1) ^∝ H_n x^n H_n = Σ_(k=1) ^(k=n) (1/k) .

$${find}\:\:{the}\:{radius}\:{of}\:{convergence}\:{for}\:{the}\:{serie}\:\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \:{H}_{{n}} \:{x}^{{n}} \\ $$$${H}_{{n}} \:\:=\:\:\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:\:\frac{\mathrm{1}}{{k}}\:. \\ $$

Question Number 26223 Answers: 1 Comments: 1

let put ξ(x)= Σ_(n=1) ^∝ (1/n^x ) with x>1 and δ(x) =Σ_(n=1) ^∝ (((−1)^n )/n^x ) find a relation between ξ(x) and δ(x).

$${let}\:{put}\:\xi\left({x}\right)=\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\frac{\mathrm{1}}{{n}^{{x}} }\:\:{with}\:{x}>\mathrm{1} \\ $$$${and}\:\:\delta\left({x}\right)\:\:=\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} }\:\:\:{find}\:{a}\:{relation} \\ $$$${between}\:\xi\left({x}\right)\:{and}\:\delta\left({x}\right). \\ $$

Question Number 26098 Answers: 0 Comments: 0

Question Number 26090 Answers: 1 Comments: 0

∫((asin^3 θ+bcos^3 θ)/(sin^2 θ.cos^2 θ))dθ

$$\int\frac{{a}\mathrm{sin}\:^{\mathrm{3}} \theta+{b}\mathrm{cos}\:^{\mathrm{3}} \theta}{\mathrm{sin}^{\mathrm{2}} \:\theta.\mathrm{cos}^{\mathrm{2}} \:\theta}{d}\theta \\ $$

Question Number 26087 Answers: 0 Comments: 2

Given f(x) = (1 − x + x^2 − x^3 + ... − x^(2015) + x^(2016) )^2 Find the sum of all odd coeffisiens! Ex. f(x) = (x^2 + x + 1)^2 = 1x^4 + 2x^3 + 3x^2 + 2x + 1 The sum of odd coeffisien is 1 + 3 = 4

$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\left(\mathrm{1}\:−\:{x}\:+\:{x}^{\mathrm{2}} \:−\:{x}^{\mathrm{3}} \:+\:...\:−\:{x}^{\mathrm{2015}} \:+\:{x}^{\mathrm{2016}} \right)^{\mathrm{2}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{odd}\:\mathrm{coeffisiens}! \\ $$$$ \\ $$$$\mathrm{Ex}.\:{f}\left({x}\right)\:=\:\left({x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{1}{x}^{\mathrm{4}} \:+\:\mathrm{2}{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1} \\ $$$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{coeffisien}\:\mathrm{is}\:\mathrm{1}\:+\:\mathrm{3}\:=\:\mathrm{4} \\ $$

Question Number 26078 Answers: 1 Comments: 0

x^2 −x−42 factorise

$$\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\mathrm{42}\:\mathrm{factorise} \\ $$

Question Number 26073 Answers: 0 Comments: 2

solve the differential equation(D^2 +2D+1)y=x^2 +2x+1

$$ \\ $$$${solve}\:{the}\:{differential}\:{equation}\left({D}^{\mathrm{2}} +\mathrm{2}{D}+\mathrm{1}\right){y}={x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1} \\ $$

Question Number 26067 Answers: 2 Comments: 0

Find the value of ((2 + 3^2 )/(1! + 2! + 3! + 4!)) + ((3 + 4^2 )/(2! + 3! + 4! + 5!)) + ... + ((2013 + 2014^2 )/(2012! + 2013! + 2014! + 2015!))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{2}\:+\:\mathrm{3}^{\mathrm{2}} }{\mathrm{1}!\:+\:\mathrm{2}!\:+\:\mathrm{3}!\:+\:\mathrm{4}!}\:+\:\frac{\mathrm{3}\:+\:\mathrm{4}^{\mathrm{2}} }{\mathrm{2}!\:+\:\mathrm{3}!\:+\:\mathrm{4}!\:+\:\mathrm{5}!}\:+\:...\:+\:\frac{\mathrm{2013}\:+\:\mathrm{2014}^{\mathrm{2}} }{\mathrm{2012}!\:+\:\mathrm{2013}!\:+\:\mathrm{2014}!\:+\:\mathrm{2015}!} \\ $$

Question Number 26109 Answers: 0 Comments: 1

let s give S_n = Σ_(k=1) ^(k=n) k^(−2 ) . (k+1)^(−2) find lim_(n−>∝) S_n .

$${let}\:{s}\:{give}\:\:{S}_{{n}} \:=\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:{k}^{−\mathrm{2}\:} .\:\left({k}+\mathrm{1}\right)^{−\mathrm{2}} \\ $$$${find}\:\:{lim}_{{n}−>\propto} \:\:{S}_{{n}} \:\:. \\ $$

Question Number 26059 Answers: 0 Comments: 0

let s give n from N find the value of ∫_0 ^1 (1 +x^2 )^(n/2) sin(n arctan(x))dx

$${let}\:{s}\:{give}\:{n}\:{from}\:{N}\:\:{find}\:{the}\:{value}\:{of} \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{1}\:+{x}^{\mathrm{2}} \:\right)^{{n}/\mathrm{2}} \:{sin}\left({n}\:{arctan}\left({x}\right)\right){dx} \\ $$

Question Number 26058 Answers: 0 Comments: 0

Question Number 26057 Answers: 0 Comments: 1

find a integral form of L( e^(−x^2 ) ) L(f) means laplace transform of f .

$${find}\:{a}\:{integral}\:{form}\:{of}\:\:{L}\left(\:\:{e}^{−{x}^{\mathrm{2}} } \:\right) \\ $$$${L}\left({f}\right)\:{means}\:{laplace}\:{transform}\:{of}\:{f}\:. \\ $$

Question Number 26055 Answers: 1 Comments: 0

calculate ∫_0 ^1 (1+t^2 )^(1/2) dt

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{2}} {dt} \\ $$

Question Number 26054 Answers: 0 Comments: 0

if (1+cos(x))^(−1) = a_0 /_2 +Σ_(n=1) ^(n=∝) a_n cos(nx)) find a_0 and a_n ...you can use fourier series.

$$\left.{if}\:\:\left(\mathrm{1}+{cos}\left({x}\right)\right)^{−\mathrm{1}} \:=\:{a}_{\mathrm{0}} /_{\mathrm{2}} \:\:+\sum_{{n}=\mathrm{1}} ^{{n}=\propto} \:{a}_{{n}} \:{cos}\left({nx}\right)\right) \\ $$$${find}\:{a}_{\mathrm{0}} \:{and}\:\:{a}_{{n}} ...{you}\:{can}\:{use}\:{fourier} \\ $$$${series}. \\ $$$$ \\ $$

Question Number 26053 Answers: 1 Comments: 0

If x^2 + 9x + 2 = 0 and x^2 + kx + 5 = 0 have a common root, show that 2k^2 + 63k − 414 = 0 , hence find the value of k such that k > 9.3

$$\mathrm{If}\:\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{9x}\:+\:\mathrm{2}\:=\:\mathrm{0}\:\:\mathrm{and}\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{kx}\:+\:\mathrm{5}\:=\:\mathrm{0}\:\:\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{root},\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\mathrm{2k}^{\mathrm{2}} \:+\:\mathrm{63k}\:−\:\mathrm{414}\:=\:\mathrm{0}\:,\:\:\mathrm{hence}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{k}\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{k}\:>\:\mathrm{9}.\mathrm{3} \\ $$

Question Number 26050 Answers: 1 Comments: 0

Question Number 26047 Answers: 2 Comments: 6

Question Number 26046 Answers: 1 Comments: 0

x^3 + (1/x^3 )=18 find the valu of x+(1/x)

$${x}^{\mathrm{3}} +\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }=\mathrm{18}\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{valu}\:\mathrm{of}\:{x}+\frac{\mathrm{1}}{{x}} \\ $$

Question Number 26042 Answers: 0 Comments: 2

ax^2 −bx=0

$${ax}^{\mathrm{2}} −{bx}=\mathrm{0} \\ $$

Question Number 26040 Answers: 0 Comments: 0

Question Number 26110 Answers: 0 Comments: 0

let s give n from N find the value of ∫_0 ^π ((sin(nx))/(sinx)) dx .

$${let}\:{s}\:{give}\:{n}\:{from}\:{N}\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}\left({nx}\right)}{{sinx}}\:{dx}\:. \\ $$

Question Number 26036 Answers: 1 Comments: 0

Find the area of a square if the sum of the diagonals is 100 cm.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{if}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{diagonals}\:\mathrm{is}\:\mathrm{100}\:\mathrm{cm}. \\ $$

Question Number 26031 Answers: 0 Comments: 0

calculate the number of protons which would have a charge of one coulomb charge(Proton Charge=1.6×10^(−19) C).

$${calculate}\:{the}\:{number}\:{of}\:{protons}\:{which}\:{would}\:{have}\:{a}\:{charge}\:{of}\:{one}\:{coulomb}\:{charge}\left({Proton}\:{Charge}=\mathrm{1}.\mathrm{6}×\mathrm{10}^{−\mathrm{19}} {C}\right). \\ $$

Question Number 26024 Answers: 0 Comments: 0

e give a element from]0.∝[ find the value of ∫_0 ^∞ cos( ax^2 ) and ∫_0 ^∞ sin( ax^2 )dx.

$$\left.{e}\:{give}\:{a}\:{element}\:{from}\right]\mathrm{0}.\propto\left[\:\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left(\:{ax}^{\mathrm{2}} \right)\right. \\ $$$${and}\:\:\int_{\mathrm{0}} ^{\infty} \:{sin}\left(\:{ax}^{\mathrm{2}} \right){dx}. \\ $$

Question Number 26023 Answers: 0 Comments: 0

answer to question25980 key of slution we develop the foction f(x) = sin(px) at fourier serie((f 2π periodic) f(x)= Σ_(n=1) ^(n=∝) a_n sin(nx) and a_n = 2/T ∫_([T]) sin(px)sin(nx)dx (T=2π) a_n =2π^(−1) ∫_0 ^π sin(px)sin(nx)dx−>a_n = (−1)^n sin(pπ).2n π^(−1) (n^2 − p^2 )^(−1) −−>sin(px)= 2 sin(pπ).π^(−1) Σ_(n=1) ^∝ n(−1)^(n−1) (n^2 −p^2 )^(−1) sin(nx) = 2π^(−1) sin(pπ)((1^2 −p^2 )^(−1) sin (x) −2(2^2 −p^2 )^(−1) sin(2x)+...)

$${answer}\:{to}\:{question}\mathrm{25980}\:{key}\:{of}\:{slution}\:{we}\:{develop}\:\:{the} \\ $$$${foction}\:{f}\left({x}\right)\:=\:{sin}\left({px}\right)\:{at}\:{fourier}\:{serie}\left(\left({f}\:\mathrm{2}\pi\:{periodic}\right)\right. \\ $$$${f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{{n}=\propto} \:{a}_{{n}} {sin}\left({nx}\right)\:{and}\:\:{a}_{{n}} =\:\mathrm{2}/{T}\:\int_{\left[{T}\right]} {sin}\left({px}\right){sin}\left({nx}\right){dx}\:\:\:\left({T}=\mathrm{2}\pi\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{2}\pi^{−\mathrm{1}} \:\int_{\mathrm{0}} ^{\pi} {sin}\left({px}\right){sin}\left({nx}\right){dx}−>{a}_{{n}} =\:\left(−\mathrm{1}\right)^{{n}} \:{sin}\left({p}\pi\right).\mathrm{2}{n}\:\pi^{−\mathrm{1}} \left({n}^{\mathrm{2}} \:−\:{p}^{\mathrm{2}} \right)^{−\mathrm{1}} \\ $$$$−−>{sin}\left({px}\right)=\:\mathrm{2}\:{sin}\left({p}\pi\right).\pi^{−\mathrm{1}} \:\sum_{{n}=\mathrm{1}} ^{\propto} \:{n}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}^{\mathrm{2}} −{p}^{\mathrm{2}} \right)^{−\mathrm{1}} \:\:{sin}\left({nx}\right) \\ $$$$=\:\mathrm{2}\pi^{−\mathrm{1}} \:{sin}\left({p}\pi\right)\left(\left(\mathrm{1}^{\mathrm{2}} −{p}^{\mathrm{2}} \right)^{−\mathrm{1}} \:{sin}\:\left({x}\right)\:−\mathrm{2}\left(\mathrm{2}^{\mathrm{2}} −{p}^{\mathrm{2}} \right)^{−\mathrm{1}} {sin}\left(\mathrm{2}{x}\right)+...\right) \\ $$

Question Number 26021 Answers: 1 Comments: 1

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