Question and Answers Forum

let give f_α (t)=cos(αt) 2π periodic with t ∈[−π,π]and α∈ R−Z 1) developp f_α at fourier serie and prove that cotan(απ)= (1/(απ)) +Σ_(n=1) ^∞ ((2α)/(π(α^2 −n^2 ))) 2)let x∈]0,π[ ant g(t)=cotant −(1/t) if t∈]0,x]andg(0)=0 prove that g is continue in[0,x] and find ∫_0 ^x g(t)dt 3)prove that ∀ t∈[0,x] g(t)=2t Σ_(n=1) ^∞ (1/(t^2 −n^2 π^(24) )) 4) chow that Π_(n=1) ^∞ (1−(x^2 /(n^2 π^2 )))= ((sinx)/x) and for x∈]−π,π[ sinx=x Π_(n=1) ^∞ (1−(x^2 /(n^2 π^2 ))) .

$${let}\:{give}\:{f}_{\alpha} \left({t}\right)={cos}\left(\alpha{t}\right)\:\:\mathrm{2}\pi\:{periodic}\:{with}\:{t}\:\in\left[−\pi,\pi\right]{and} \\ $$$$\alpha\in\:{R}−{Z} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}_{\alpha} \:\:{at}\:{fourier}\:{serie}\:{and}\:{prove}\:{that} \\ $$$${cotan}\left(\alpha\pi\right)=\:\frac{\mathrm{1}}{\alpha\pi}\:\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{2}\alpha}{\pi\left(\alpha^{\mathrm{2}} −{n}^{\mathrm{2}} \right)} \\ $$$$\left.\mathrm{2}\left.\right)\left.{let}\:{x}\in\right]\mathrm{0},\pi\left[\:{ant}\:{g}\left({t}\right)={cotant}\:−\frac{\mathrm{1}}{{t}}\:\:{if}\:{t}\in\right]\mathrm{0},{x}\right]{andg}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:{g}\:{is}\:{continue}\:{in}\left[\mathrm{0},{x}\right]\:{and}\:{find}\:\int_{\mathrm{0}} ^{{x}} {g}\left({t}\right){dt} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:\forall\:{t}\in\left[\mathrm{0},{x}\right]\:{g}\left({t}\right)=\mathrm{2}{t}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{t}^{\mathrm{2}} −{n}^{\mathrm{2}} \pi^{\mathrm{24}} } \\ $$$$\left.\mathrm{4}\left.\right)\:{chow}\:{that}\:\:\prod_{{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\right)=\:\frac{{sinx}}{{x}}\:\:{and}\:{for}\:{x}\in\right]−\pi,\pi\left[\right. \\ $$$${sinx}={x}\:\prod_{{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\right)\:\:. \\ $$

Question Number 29842 Answers: 0 Comments: 1

prove that ∀ x∈]0,1[ (1/(Γ(x).Γ(1−x)))=x Π_(n=1) ^∞ (1−(x^2 /n^2 )).

$$\left.{prove}\:{that}\:\forall\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\:\:\:\:\:\frac{\mathrm{1}}{\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)}={x}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}} }\right).\right. \\ $$

Question Number 29841 Answers: 0 Comments: 0

prove that ∀ x∈]0,1[ Γ(x).Γ(1−x)= (π/(sin(πx))) (compliments formula).

$$\left.{prove}\:{that}\:\forall\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\:\:\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)=\:\frac{\pi}{{sin}\left(\pi{x}\right)}\right. \\ $$$$\left({compliments}\:{formula}\right). \\ $$

Question Number 29885 Answers: 1 Comments: 2

∫(1/(1+sinx))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{1}+\mathrm{sin}{x}}{dx}=? \\ $$

Question Number 29839 Answers: 0 Comments: 0

find Π_(n=1) ^∞ (1−(x^2 /(n^2 π^2 ))).

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

Question Number 29838 Answers: 0 Comments: 0

find Π_(n=1) ^∞ (1−(1/(4n^2 ))).

$${find}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}{n}^{\mathrm{2}} }\right). \\ $$

Question Number 29837 Answers: 0 Comments: 0

let give T_n (x)=cos(n arcosx) with x∈[−1,1] 1) prove that T_n is a polynomial and T_n ∈Z[x] 2)calculate T_1 , T_2 , T_3 ,and T_4 3) prove that T_(n+2) (x)=2x T_(n+1) (x)−T_n (x) 4)find the roots of T_n and factorize T_n (x).

$${let}\:{give}\:\:{T}_{{n}} \left({x}\right)={cos}\left({n}\:{arcosx}\right)\:{with}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{T}_{{n}} \:{is}\:{a}\:{polynomial}\:{and}\:{T}_{{n}} \in{Z}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right){calculate}\:{T}_{\mathrm{1}} ,\:{T}_{\mathrm{2}} ,\:{T}_{\mathrm{3}} ,{and}\:{T}_{\mathrm{4}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{T}_{{n}+\mathrm{2}} \left({x}\right)=\mathrm{2}{x}\:{T}_{{n}+\mathrm{1}} \left({x}\right)−{T}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{4}\right){find}\:{the}\:{roots}\:{of}\:{T}_{{n}} \:{and}\:{factorize}\:{T}_{{n}} \left({x}\right). \\ $$

Question Number 29836 Answers: 0 Comments: 0

let give u_n = Σ_(q=1) ^n (1/(n^2 +q)) find lim_(n→+∞) (1−nu_n )n.

$${let}\:{give}\:{u}_{{n}} =\:\sum_{{q}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{q}}\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \left(\mathrm{1}−{nu}_{{n}} \right){n}. \\ $$

Question Number 29835 Answers: 0 Comments: 0

let give f(x)=−x +2 +((√(x+1))/x) 1) study the variation of and give the graph C_f 2)give the equation of tangent at C_f in point A(1,f(1))

$${let}\:{give}\:{f}\left({x}\right)=−{x}\:+\mathrm{2}\:+\frac{\sqrt{{x}+\mathrm{1}}}{{x}} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{variation}\:{of}\:{and}\:{give}\:{the}\:{graph}\:{C}_{{f}} \\ $$$$\left.\mathrm{2}\right){give}\:{the}\:{equation}\:{of}\:{tangent}\:{at}\:{C}_{{f}} \:{in}\:{point}\:{A}\left(\mathrm{1},{f}\left(\mathrm{1}\right)\right) \\ $$

Question Number 29834 Answers: 0 Comments: 1

find (1/(cos^4 ((π/9)))) +(1/(cos^4 (((3π)/9)))) + (1/(cos^4 (((5π)/9)))) +(1/(cos^4 (((7π)/9)))) .

$${find}\:\:\:\:\frac{\mathrm{1}}{{cos}^{\mathrm{4}} \left(\frac{\pi}{\mathrm{9}}\right)}\:+\frac{\mathrm{1}}{{cos}^{\mathrm{4}} \left(\frac{\mathrm{3}\pi}{\mathrm{9}}\right)}\:+\:\frac{\mathrm{1}}{{cos}^{\mathrm{4}} \left(\frac{\mathrm{5}\pi}{\mathrm{9}}\right)}\:+\frac{\mathrm{1}}{{cos}^{\mathrm{4}} \left(\frac{\mathrm{7}\pi}{\mathrm{9}}\right)}\:. \\ $$

Question Number 29833 Answers: 1 Comments: 0

find cos^4 ((π/8)) +cos^4 (((3π)/8)) +cos^4 (((5π)/8)) +cos^4 (((7π)/8)).

$${find}\:\:{cos}^{\mathrm{4}} \left(\frac{\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right). \\ $$

Question Number 29832 Answers: 0 Comments: 0

p is a polynomial having n roots x_i with x_i ≠x_j for i≠j prove that Σ_(i=1) ^n ((p^(′′) (x_i ))/(p^′ (x_i )))=0

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{n}\:{roots}\:{x}_{{i}} \:\:{with}\:{x}_{{i}} \neq{x}_{{j}} \:{for}\:{i}\neq{j} \\ $$$${prove}\:{that}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\frac{{p}^{''} \left({x}_{{i}} \right)}{{p}^{'} \left({x}_{{i}} \right)}=\mathrm{0} \\ $$

Question Number 29821 Answers: 1 Comments: 3

((sin 16x)/(sin x)) ?pls help.

$$\frac{\mathrm{sin}\:\mathrm{16x}}{\mathrm{sin}\:\mathrm{x}}\:\:\:\:\:\:?\mathrm{pls}\:\mathrm{help}. \\ $$

Question Number 29820 Answers: 1 Comments: 3

Question Number 29805 Answers: 0 Comments: 1

f(x)=(x+a_1 )(x+a_2 )(x+a_3 )...(x+a_n ) find the coefficient of term x^k (0≤k≤n)

$${f}\left({x}\right)=\left({x}+{a}_{\mathrm{1}} \right)\left({x}+{a}_{\mathrm{2}} \right)\left({x}+{a}_{\mathrm{3}} \right)...\left({x}+{a}_{{n}} \right) \\ $$$${find}\:{the}\:{coefficient}\:{of}\:{term}\:{x}^{{k}} \:\left(\mathrm{0}\leqslant{k}\leqslant{n}\right) \\ $$

Question Number 29818 Answers: 1 Comments: 0

4, 8, 16, 31, 57, 99, 163, T_8 ,T_9 , .... Find T_8 , T_9 .

$$\mathrm{4},\:\mathrm{8},\:\mathrm{16},\:\mathrm{31},\:\mathrm{57},\:\mathrm{99},\:\mathrm{163},\:{T}_{\mathrm{8}} \:,{T}_{\mathrm{9}} \:,\:.... \\ $$$${Find}\:{T}_{\mathrm{8}} \:,\:{T}_{\mathrm{9}} \:. \\ $$

Question Number 29794 Answers: 0 Comments: 9

Fluids:

$${Fluids}: \\ $$

Question Number 29831 Answers: 0 Comments: 0

let give f(x)= (1/(1+x^2 )) 1)prove that prove that f^((n)) (x)=((p_n (x))/((1+x^2 )^(n+1) )) with p_n is a polynomial 2) prove that p_(n+1) (x)=(1+x^2 )p_n ^′ (x) −2(n+1)p_n (x) 3) calculate p_0 (x) ,p_1 (x) ,p_2 (x) ,p_3 (x) .

$$\left.{let}\:{give}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\right){prove}\:{that}\:\:\:{prove}\:{that} \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\frac{{p}_{{n}} \left({x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}+\mathrm{1}} }\:{with}\:{p}_{{n}} {is}\:{a}\:{polynomial} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{p}_{{n}+\mathrm{1}} \left({x}\right)=\left(\mathrm{1}+{x}^{\mathrm{2}} \right){p}_{{n}} ^{'} \left({x}\right)\:−\mathrm{2}\left({n}+\mathrm{1}\right){p}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{p}_{\mathrm{0}} \left({x}\right)\:,{p}_{\mathrm{1}} \left({x}\right)\:,{p}_{\mathrm{2}} \left({x}\right)\:\:,{p}_{\mathrm{3}} \left({x}\right)\:\:. \\ $$

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