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Question Number 32995    Answers: 0   Comments: 1

let u_0 =a , u_1 =b and u_(n+2) =(1/2)(u_n +u_(n+1) ) 1) find u_n interms of n 2) find lim_(n→∞) u_n if a=0

$${let}\:{u}_{\mathrm{0}} ={a}\:,\:{u}_{\mathrm{1}} ={b}\:{and}\:{u}_{{n}+\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}} \:+{u}_{{n}+\mathrm{1}} \right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{u}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} \:\:{if}\:{a}=\mathrm{0} \\ $$$$ \\ $$

Question Number 32994    Answers: 0   Comments: 1

find ∫_0 ^∞ ((1−cos(λx))/x^2 ) dx with λ>0 .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}−{cos}\left(\lambda{x}\right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:\lambda>\mathrm{0}\:. \\ $$

Question Number 32993    Answers: 0   Comments: 0

calculate ∫_0 ^∞ e^(−λx) ((sinx)/(√x)) dx wih λ>0 .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\lambda{x}} \:\frac{{sinx}}{\sqrt{{x}}}\:{dx}\:\:{wih}\:\lambda>\mathrm{0}\:. \\ $$

Question Number 32992    Answers: 0   Comments: 0

find lim_(x→0) ((tan(sinx) −sin(tanx))/x^2 ) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{tan}\left({sinx}\right)\:−{sin}\left({tanx}\right)}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32991    Answers: 1   Comments: 1

calculate lim_(x→0) ((ln(1+x) −x)/x^2 ) .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{ln}\left(\mathrm{1}+{x}\right)\:−{x}}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32990    Answers: 1   Comments: 1

find lim_(x→0^+ ) x e^(−lnx) .

$$\:\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{x}\:{e}^{−{lnx}} \:\:. \\ $$

Question Number 32989    Answers: 0   Comments: 0

let give f(x)=(((√(2+x)) −(√(2−x)) )/x) find f^(−1) (x) and calculate (f^(−1) (x))^′ .

$${let}\:{give}\:{f}\left({x}\right)=\frac{\sqrt{\mathrm{2}+{x}}\:−\sqrt{\mathrm{2}−{x}}\:}{{x}} \\ $$$${find}\:{f}^{−\mathrm{1}} \left({x}\right)\:{and}\:{calculate}\:\left({f}^{−\mathrm{1}} \left({x}\right)\right)^{'} \:\:. \\ $$

Question Number 32987    Answers: 1   Comments: 0

If 1=1 2=4 3=10 4=20 5=?

$${If}\:\mathrm{1}=\mathrm{1} \\ $$$$\mathrm{2}=\mathrm{4} \\ $$$$\mathrm{3}=\mathrm{10}\: \\ $$$$\mathrm{4}=\mathrm{20} \\ $$$$\mathrm{5}=? \\ $$

Question Number 32985    Answers: 1   Comments: 0

Question Number 32980    Answers: 1   Comments: 0

(d/dx)( determinant ((((x^2 +2)^(x^3 +3) ),2^x ,(cosx^x )),((log_2^(x+1) (x^2 +x^(2x) +3^ )),(xlnx),(sin^(−1) tanx)),(3,π^(sinhx) ,e^x )))=?

$$ \\ $$$$ \\ $$$$\frac{\mathrm{d}}{\mathrm{dx}}\left(\begin{vmatrix}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{x}^{\mathrm{3}} +\mathrm{3}} }&{\mathrm{2}^{\mathrm{x}} }&{\mathrm{cosx}^{\mathrm{x}} }\\{\mathrm{log}_{\mathrm{2}^{\mathrm{x}+\mathrm{1}} } \left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2x}} +\overset{} {\mathrm{3}}\right)}&{\mathrm{xlnx}}&{\mathrm{sin}^{−\mathrm{1}} \mathrm{tanx}}\\{\mathrm{3}}&{\pi^{\mathrm{sinhx}} }&{\mathrm{e}^{\mathrm{x}} }\end{vmatrix}\right)=? \\ $$$$ \\ $$

Question Number 33152    Answers: 0   Comments: 1

it is given that (1/n)Σ_(r=1) ^n x^r =2 and (√((1/n)Σ_(r=1) ^n (x_r )^2 −(1/n^2 )(Σ_(r=1) ^n )^2 ))= 3 determine in terms of n the value of. Σ_(r=1) ^n (x_r +1)^2

$${it}\:{is}\:{given}\:{that} \\ $$$$\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{x}^{{r}} =\mathrm{2}\:{and}\:\sqrt{\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left({x}_{{r}} \right)^{\mathrm{2}} −\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\right)^{\mathrm{2}} }=\:\mathrm{3} \\ $$$${determine}\:{in}\:{terms}\:{of}\:{n}\:{the}\:{value} \\ $$$${of}. \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left({x}_{{r}} +\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 32977    Answers: 1   Comments: 0

Prove that ^n C_r +^n C_(r+1) =^(n+1) C_(r+1)

$${Prove}\:{that}\:\:^{{n}} {C}_{{r}} \:\:+\:^{{n}} {C}_{{r}+\mathrm{1}} \:=\:^{{n}+\mathrm{1}} {C}_{{r}+\mathrm{1}} \\ $$$$ \\ $$

Question Number 33125    Answers: 0   Comments: 0

let give u_n = ∫_0 ^π ((cos(nx)dx)/(1−2λcosx +λ^2 )) 1) prove that λ u_(n+2) −(1+λ^2 )u_(n+1) +λ u_n =0 2) ptove that Σ u_n is convergent and find its sum

$${let}\:{give}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{cos}\left({nx}\right){dx}}{\mathrm{1}−\mathrm{2}\lambda{cosx}\:+\lambda^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\lambda\:{u}_{{n}+\mathrm{2}} \:−\left(\mathrm{1}+\lambda^{\mathrm{2}} \right){u}_{{n}+\mathrm{1}} \:+\lambda\:{u}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{ptove}\:{that}\:\Sigma\:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{sum} \\ $$

Question Number 33124    Answers: 0   Comments: 1

find Σ_(n=1) ^∞ (1/(n(n+1)2^(n−1) )) .

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

Question Number 32971    Answers: 1   Comments: 0

If siny=xsin(a+y) show that (dy/dx)=((sina)/(1−2xcosa+x^2 ))

$${If}\:{siny}={xsin}\left({a}+{y}\right)\:{show}\:{that} \\ $$$$\frac{{dy}}{{dx}}=\frac{{sina}}{\mathrm{1}−\mathrm{2}{xcosa}+{x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 32968    Answers: 1   Comments: 0

Question Number 32958    Answers: 0   Comments: 1

Question Number 32954    Answers: 1   Comments: 0

log_2 x×log_4 x×log_(16) x=7 help

$$\:\boldsymbol{\mathrm{log}}_{\mathrm{2}} \boldsymbol{{x}}×\boldsymbol{\mathrm{log}}_{\mathrm{4}} \boldsymbol{{x}}×\boldsymbol{\mathrm{log}}_{\mathrm{16}} \boldsymbol{{x}}=\mathrm{7} \\ $$$$\:\boldsymbol{{help}} \\ $$

Question Number 32951    Answers: 2   Comments: 1

Evaluate ∫((x^4 +1)/(x^6 +1))dx [W.B.H.S 2018]

$${Evaluate} \\ $$$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx}\:\:\:\:\:\left[{W}.{B}.{H}.{S}\:\mathrm{2018}\right] \\ $$

Question Number 32947    Answers: 1   Comments: 0

Question Number 32946    Answers: 1   Comments: 0

Question Number 32945    Answers: 1   Comments: 0

Question Number 32939    Answers: 1   Comments: 1

1) study the convergence of ∫_0 ^1 (x^p /(1+x)) dx 2) find lim_(p→∞) ∫_0 ^1 (x^p /(1+x))dx .

$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{{p}} }{\mathrm{1}+{x}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{p}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{{p}} }{\mathrm{1}+{x}}{dx}\:. \\ $$

Question Number 32938    Answers: 0   Comments: 2

let 0<θ<π find Σ_(n=1) ^∞ ((cos(nθ))/n) 2) find Σ_(n=1) ^∞ ((sin(nθ))/n)

$${let}\:\mathrm{0}<\theta<\pi\:\:{find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{cos}\left({n}\theta\right)}{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({n}\theta\right)}{{n}} \\ $$

Question Number 32937    Answers: 0   Comments: 0

let give ∣x∣<1 prove that Σ_(n=1) ^∞ ((sin(nθ))/n) x^n =arctan( ((xsinθ)/(1−xcosθ))) .

$${let}\:{give}\:\mid{x}\mid<\mathrm{1}\:\:{prove}\:{that} \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({n}\theta\right)}{{n}}\:{x}^{{n}} \:={arctan}\left(\:\frac{{xsin}\theta}{\mathrm{1}−{xcos}\theta}\right)\:. \\ $$

Question Number 32936    Answers: 0   Comments: 0

let give ∣x∣<1 prove that Σ_(n=1) ^∞ ((cos(nθ))/n) x^n =−(1/2)ln(1−2xcosθ+x^2 ) .

$${let}\:{give}\:\mid{x}\mid<\mathrm{1}\:{prove}\:{that} \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{cos}\left({n}\theta\right)}{{n}}\:{x}^{{n}} \:=−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}−\mathrm{2}{xcos}\theta+{x}^{\mathrm{2}} \right)\:. \\ $$

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