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Question Number 36751 Answers: 0 Comments: 0

let f(x)= Σ_(n=1) ^∞ x^n^2 with x∈]−1,1[ prove that f(x) ∼ ((√π)/(2(√(−ln(x))))) (x →1^− )

$$\left.{let}\:\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{x}^{{n}^{\mathrm{2}} } \:\:\:{with}\:\:{x}\in\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$${prove}\:{that}\:\:{f}\left({x}\right)\:\sim\:\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{−{ln}\left({x}\right)}}\:\left({x}\:\rightarrow\mathrm{1}^{−} \right) \\ $$

Question Number 36750 Answers: 0 Comments: 0

let f(t)=Σ_(n≥1) (−1)^n ln{1+ (t^2 /(n(1+t^2 )))} 1) study the simple and uniform convergence of Σ f_n 2)study the continuity of f 3) prove that lim_(t→+∞) f(t)=ln((2/π)) .

$${let}\:{f}\left({t}\right)=\sum_{{n}\geqslant\mathrm{1}} \:\left(−\mathrm{1}\right)^{{n}} {ln}\left\{\mathrm{1}+\:\frac{{t}^{\mathrm{2}} }{{n}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\right\} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{simple}\:\:{and}\:{uniform}\:{convergence} \\ $$$${of}\:\Sigma\:{f}_{{n}} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continuity}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{t}\rightarrow+\infty} \:{f}\left({t}\right)={ln}\left(\frac{\mathrm{2}}{\pi}\right)\:. \\ $$

Question Number 36748 Answers: 0 Comments: 0

let f(x)= Σ_(n=1) ^∞ (((−1)^(n−1) )/(ln(nx))) 1) give D_f and study f on]1,+∞[ 2)study the continjity of f and calculate lim _(x→1) f(x) and lim_(x→+∞) f(x). 3) prove that f is C^1 on ]1,+∞[ .

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{ln}\left({nx}\right)} \\ $$$$\left.\mathrm{1}\left.\right)\:{give}\:{D}_{{f}} \:\:{and}\:{study}\:{f}\:{on}\right]\mathrm{1},+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continjity}\:{of}\:{f}\:{and}\:{calculate} \\ $$$${lim}\:_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]\mathrm{1},+\infty\left[\:.\right. \\ $$

Question Number 36747 Answers: 0 Comments: 1

let f(x)= Σ_(n=1) ^∞ ((sin(nx))/n) x^n 1) prove that f is C^1 on ]−1,1[ 2)calculate f^′ (x) and prove that f(x)=arctan( ((xsinx)/(1−x cosx)))

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}}\:{x}^{{n}} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({x}\right)\:{and}\:{prove}\:{that} \\ $$$${f}\left({x}\right)={arctan}\left(\:\frac{{xsinx}}{\mathrm{1}−{x}\:{cosx}}\right) \\ $$

Question Number 36746 Answers: 0 Comments: 0

solve the d.e. y^(′′) (t) +y(t)=Σ_(n=0) ^N a_n cos(nt)

$${solve}\:{the}\:{d}.{e}.\:{y}^{''} \left({t}\right)\:+{y}\left({t}\right)=\sum_{{n}=\mathrm{0}} ^{{N}} \:{a}_{{n}} {cos}\left({nt}\right) \\ $$

Question Number 36745 Answers: 0 Comments: 0

solve the d.e y^(′′) (t) +y(t)=cos(nt)

$${solve}\:{the}\:{d}.{e}\:\:{y}^{''} \left({t}\right)\:+{y}\left({t}\right)={cos}\left({nt}\right) \\ $$

Question Number 36744 Answers: 0 Comments: 0

let f(x)=Σ_(n=1) ^∞ (1/n) cos^n (x)sin(nx) 1)prove the convergence of this serie 2)prove that f is C^2 on R −{kπ,k∈Z}and calculate f^′ (x) 3) give a exprrssion of f.

$${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}}\:{cos}^{{n}} \left({x}\right){sin}\left({nx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{2}} \:{on}\:{R}\:−\left\{{k}\pi,{k}\in{Z}\right\}{and} \\ $$$${calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{a}\:{exprrssion}\:{of}\:{f}. \\ $$

Question Number 36742 Answers: 0 Comments: 1

study the convergence of Σ_(n=1) ^∞ (−1)^n ln(1+ (1/(n(1+x)))).

$${study}\:{the}\:{convergence}\:{of}\: \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} {ln}\left(\mathrm{1}+\:\frac{\mathrm{1}}{{n}\left(\mathrm{1}+{x}\right)}\right). \\ $$

Question Number 36741 Answers: 1 Comments: 1

calculate S(x)=Σ_(n=0) ^∞ ((sin(nx))/(n!))

$${calculate}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{n}!} \\ $$

Question Number 36739 Answers: 0 Comments: 3

If x = a(1 − cosθ)i + asinθ j find the resultant of x in its simplest form.

$$\mathrm{If}\:\:\:\mathrm{x}\:=\:\mathrm{a}\left(\mathrm{1}\:−\:\mathrm{cos}\theta\right)\mathrm{i}\:\:+\:\:\mathrm{asin}\theta\:\mathrm{j} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{resultant}\:\mathrm{of}\:\mathrm{x}\:\mathrm{in}\:\mathrm{its}\:\mathrm{simplest}\:\mathrm{form}. \\ $$

Question Number 36738 Answers: 2 Comments: 3

(1) ∫(dα/((1+sin 2α)^2 ))= (2) ∫(dβ/((1+cos 2β)^2 ))= (3) ∫(dγ/((1+sin 2γ)(1+cos 2γ)))=

$$\left(\mathrm{1}\right)\:\:\:\:\:\int\frac{{d}\alpha}{\left(\mathrm{1}+\mathrm{sin}\:\mathrm{2}\alpha\right)^{\mathrm{2}} }= \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\int\frac{{d}\beta}{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\beta\right)^{\mathrm{2}} }= \\ $$$$\left(\mathrm{3}\right)\:\:\:\:\:\int\frac{{d}\gamma}{\left(\mathrm{1}+\mathrm{sin}\:\mathrm{2}\gamma\right)\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\gamma\right)}= \\ $$

Question Number 36775 Answers: 0 Comments: 0

find the interval of convergence of Σ_(n=1) ^(+∞) (((x−2)^2 )/n)

$$\:{find}\:{the}\:{interval}\:{of}\:{convergence} \\ $$$$\:{of}\:\:\:\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\left({x}−\mathrm{2}\right)^{\mathrm{2}} }{{n}} \\ $$

Question Number 36737 Answers: 0 Comments: 1

let g(θ) =∫_0 ^1 ln( 1−e^(iθ) x^2 )dx find a simple form of g(θ) .θ from R.

$${let}\:{g}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\:\mathrm{1}−{e}^{{i}\theta} {x}^{\mathrm{2}} \right){dx} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{g}\left(\theta\right)\:.\theta\:{from}\:{R}. \\ $$

Question Number 36736 Answers: 0 Comments: 1

let f(θ) = ∫_0 ^1 ln(1−e^(iθ) x)dx find a simple form of f(θ)

$${let}\:\:{f}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}−{e}^{{i}\theta} {x}\right){dx} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left(\theta\right) \\ $$

Question Number 36728 Answers: 1 Comments: 1

the improper integral ∫_0 ^1 (dx/(√(1−x^2 ))) converges to

$${the}\:{improper}\:{integral}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{converges}\:{to} \\ $$

Question Number 36722 Answers: 1 Comments: 0

Question Number 36711 Answers: 0 Comments: 0

The first term in an AP is the 3rd term in A GP , the 5th term is 30 for the AP, and the 12th term of the GP is 64,find the sum to infinity.

$$\:\:{The}\:\:{first}\:{term}\:{in}\:{an}\:{AP}\:{is}\:{the}\: \\ $$$$\mathrm{3}{rd}\:{term}\:{in}\:{A}\:{GP}\:,\:{the}\:\mathrm{5}{th}\:{term}\:{is} \\ $$$$\mathrm{30}\:{for}\:{the}\:{AP},\:{and}\:{the}\:\mathrm{12}{th}\:{term} \\ $$$${of}\:{the}\:{GP}\:{is}\:\mathrm{64},{find}\:{the}\:{sum}\:{to} \\ $$$${infinity}. \\ $$

Question Number 36714 Answers: 0 Comments: 0

Given the matrix (((3c+1 5 )),((c c)) ) is singular find the value of c and find x if y = mx + c are all natural numbers

$${Given}\:{the}\:{matrix} \\ $$$$\begin{pmatrix}{\mathrm{3}{c}+\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{5}\:}\\{{c}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{c}}\end{pmatrix}\:{is}\:{singular}\: \\ $$$${find}\:{the}\:{value}\:{of}\:{c}\:{and}\: \\ $$$${find}\:{x}\:{if}\:{y}\:=\:{mx}\:+\:{c}\:{are}\:{all} \\ $$$${natural}\:{numbers} \\ $$$$ \\ $$

Question Number 36707 Answers: 2 Comments: 1

P=i^2 R then how (dP/P) = 2 (dI/I) ? Similarly P= (V^2 /R) then how (dP/P)=2(dV/V)?

$$\mathrm{P}=\mathrm{i}^{\mathrm{2}} \mathrm{R}\:\mathrm{then}\:\mathrm{how}\:\frac{\mathrm{dP}}{\mathrm{P}}\:=\:\mathrm{2}\:\frac{\mathrm{dI}}{\mathrm{I}}\:? \\ $$$$\mathrm{Similarly}\:\mathrm{P}=\:\frac{\mathrm{V}^{\mathrm{2}} }{\mathrm{R}}\:\mathrm{then}\:\mathrm{how}\:\frac{\mathrm{dP}}{\mathrm{P}}=\mathrm{2}\frac{\mathrm{dV}}{\mathrm{V}}? \\ $$

Question Number 36703 Answers: 2 Comments: 2

show that f(x)=sin X is derivable at every aεR

$${show}\:{that}\:{f}\left({x}\right)=\mathrm{sin}\:{X}\:{is}\:{derivable}\:{at}\:{every}\:{a}\varepsilon{R} \\ $$

Question Number 36700 Answers: 2 Comments: 0

Question Number 36699 Answers: 0 Comments: 0

Question Number 36696 Answers: 0 Comments: 0

if y=tan^(−1) x show that (1+x^2 )y_(n+2) +2(n+1)xy_(n+1) +n(n+1)y_n =0

$${if}\:{y}={tan}^{−\mathrm{1}} {x}\:{show}\:{that} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}_{{n}+\mathrm{2}} +\mathrm{2}\left({n}+\mathrm{1}\right){xy}_{{n}+\mathrm{1}} +{n}\left({n}+\mathrm{1}\right){y}_{{n}} =\mathrm{0} \\ $$

Question Number 36692 Answers: 2 Comments: 1

x^3 +y^3 =5 x^2 +y^2 =3

$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{5} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{3} \\ $$

Question Number 36691 Answers: 0 Comments: 2

Question Number 36689 Answers: 0 Comments: 1

1) find the value of ∫_0 ^1 ln(1−x^3 )dx then find the value of ∫_0 ^1 ln(1+x+x^2 )dx 2)find the value of ∫_0 ^1 ln(1+x^3 )dx then calculate ∫_0 ^1 ln(1−x +x^2 )dx

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{x}^{\mathrm{3}} \right){dx}\:{then} \\ $$$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx}\:{then}\: \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}−{x}\:+{x}^{\mathrm{2}} \right){dx} \\ $$

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