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Question Number 62455 Answers: 0 Comments: 3

Question Number 62611 Answers: 0 Comments: 2

If α and β are the roots of x^2 −(a+1)x+(1/2)(a^2 +a+1)=0 then α^2 +β^2 =_____.

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$${x}^{\mathrm{2}} −\left({a}+\mathrm{1}\right){x}+\frac{\mathrm{1}}{\mathrm{2}}\left({a}^{\mathrm{2}} +{a}+\mathrm{1}\right)=\mathrm{0}\:\mathrm{then} \\ $$$$\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\_\_\_\_\_. \\ $$

Question Number 62610 Answers: 2 Comments: 2

Find the value of x in (1/(x−1)) + (1/(x−2)) = (3/(x−3)) .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{in} \\ $$$$\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\:\frac{\mathrm{1}}{{x}−\mathrm{2}}\:=\:\frac{\mathrm{3}}{{x}−\mathrm{3}}\:\:. \\ $$

Question Number 62453 Answers: 0 Comments: 3

∫ (x/(e^x − 1))dx, for x > 0

$$\int\:\frac{\mathrm{x}}{\mathrm{e}^{\mathrm{x}} \:−\:\mathrm{1}}\mathrm{dx},\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\:\mathrm{x}\:>\:\mathrm{0} \\ $$

Question Number 62452 Answers: 1 Comments: 0

Find the remainder when 2014! is divisible by 2017

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\:\:\mathrm{2014}!\:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\:\mathrm{2017} \\ $$

Question Number 62449 Answers: 2 Comments: 1

Find the number of digit in 2^(50)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digit}\:\mathrm{in}\:\:\:\:\mathrm{2}^{\mathrm{50}} \\ $$

Question Number 62448 Answers: 1 Comments: 1

Question Number 62440 Answers: 0 Comments: 2

let h(x)= arctan(x+(1/x)) 1)calculate h^((n)) (x) and h^((n)) (1) 2)developp f(x)at integr serie at x_0 =1

$${let}\:{h}\left({x}\right)=\:{arctan}\left({x}+\frac{\mathrm{1}}{{x}}\right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{h}^{\left({n}\right)} \left({x}\right)\:{and}\:{h}^{\left({n}\right)} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\left({x}\right){at}\:{integr}\:{serie}\:{at}\:{x}_{\mathrm{0}} =\mathrm{1} \\ $$

Question Number 62439 Answers: 0 Comments: 1

sove inside Z/3Z the systeme { ((5x+7y =10)),((2x+5y =8)) :}

$${sove}\:{inside}\:{Z}/\mathrm{3}{Z}\:{the}\:{systeme} \\ $$$$\begin{cases}{\mathrm{5}{x}+\mathrm{7}{y}\:=\mathrm{10}}\\{\mathrm{2}{x}+\mathrm{5}{y}\:=\mathrm{8}}\end{cases} \\ $$$$ \\ $$

Question Number 62438 Answers: 0 Comments: 1

splve x^2 y^(′′) −(x+1)y′ =(x+1)e^(−x)

$${splve}\:{x}^{\mathrm{2}} {y}^{''} \:−\left({x}+\mathrm{1}\right){y}'\:\:\:=\left({x}+\mathrm{1}\right){e}^{−{x}} \\ $$$$ \\ $$$$ \\ $$

Question Number 62437 Answers: 0 Comments: 1

let f(x) =∫_0 ^1 ((arctan(1+xt))/(t^2 +1))dt determine a explicit form for f(x) 2)calculate ∫_0 ^1 ((arctan(1+2t))/(1+t^2 ))dt

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{1}+{xt}\right)}{{t}^{\mathrm{2}} \:+\mathrm{1}}{dt} \\ $$$${determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{1}+\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$ \\ $$

Question Number 62435 Answers: 0 Comments: 2

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

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

Question Number 62434 Answers: 0 Comments: 0

let f(x)=ch(cosx) 1)calculste f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie

$${let}\:{f}\left({x}\right)={ch}\left({cosx}\right) \\ $$$$\left.\mathrm{1}\right){calculste}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 62431 Answers: 1 Comments: 0

Question Number 62428 Answers: 0 Comments: 6

Question Number 62424 Answers: 1 Comments: 2

Question Number 62425 Answers: 0 Comments: 0

let ξ(x) =Σ_(n=1) ^∞ (1/n^x ) with x>1 1) calculate lim_(x→1^+ ) ξ(x) and lim_(x→+∞) ξ(x) 2) prove that ξ(x) =1+2^(−x) +o(2^(−x) ) (x→+∞) 3) prove that ξ is decreasing and convexe fucntion on]1,+∞[

$${let}\:\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{with}\:{x}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{1}^{+} } \:\:\xi\left({x}\right)\:\:{and}\:{lim}_{{x}\rightarrow+\infty} \:\:\:\xi\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\xi\left({x}\right)\:=\mathrm{1}+\mathrm{2}^{−{x}} \:+{o}\left(\mathrm{2}^{−{x}} \right)\:\:\:\left({x}\rightarrow+\infty\right) \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:\xi\:{is}\:{decreasing}\:{and}\:{convexe}\:{fucntion}\:{on}\right]\mathrm{1},+\infty\left[\right. \\ $$

Question Number 62420 Answers: 0 Comments: 0

let u_n (x)=(1/n^x ) −∫_n ^(n+1) (dt/t^x ) with x∈[1,2] 1)prove that 0≤ u_n (x)≤(1/n^x )−(1/((n+1)^x )) (n>0) 2)prove that Σ u_n (x)converges let γ =Σ_(n=1) ^∞ u_n (1) 3)find Σ_(n=1) ^∞ u_n (x) interms of ξ(x)and 1−x 4) prove that the converg.of Σu_n (x)is uniform prove that for x∈V(1) ξ(x) =(1/(x−1)) +γ +o(1) 5) find the value of Σ_(n=1) ^∞ (((−1)^(n−1) )/n)ln(n)

$${let}\:{u}_{{n}} \left({x}\right)=\frac{\mathrm{1}}{{n}^{{x}} }\:−\int_{{n}} ^{{n}+\mathrm{1}} \frac{{dt}}{{t}^{{x}} }\:\:{with}\:{x}\in\left[\mathrm{1},\mathrm{2}\right] \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\mathrm{0}\leqslant\:{u}_{{n}} \left({x}\right)\leqslant\frac{\mathrm{1}}{{n}^{{x}} }−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{{x}} }\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\Sigma\:{u}_{{n}} \left({x}\right){converges} \\ $$$${let}\:\gamma\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{u}_{{n}} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right){find}\:\sum_{{n}=\mathrm{1}} ^{\infty} {u}_{{n}} \left({x}\right)\:{interms}\:{of}\:\xi\left({x}\right){and} \\ $$$$\mathrm{1}−{x} \\ $$$$\left.\mathrm{4}\right)\:{prove}\:{that}\:{the}\:{converg}.{of}\:\Sigma{u}_{{n}} \left({x}\right){is} \\ $$$${uniform} \\ $$$${prove}\:{that}\:{for}\:{x}\in{V}\left(\mathrm{1}\right) \\ $$$$\xi\left({x}\right)\:=\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\gamma\:+{o}\left(\mathrm{1}\right) \\ $$$$\left.\mathrm{5}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}{ln}\left({n}\right) \\ $$

Question Number 62419 Answers: 0 Comments: 1

calculate ∫_0 ^1 (2x^2 −1)(√(x^2 −2x+5))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}{dx} \\ $$

Question Number 62418 Answers: 0 Comments: 0

calculate ∫_0 ^1 Γ(t).Γ(1−t)dt

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\Gamma\left({t}\right).\Gamma\left(\mathrm{1}−{t}\right){dt}\: \\ $$

Question Number 62417 Answers: 0 Comments: 0

prove that Γ(x).Γ(1−x) =(π/(sin(πx))) with 0<x<1

$${prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\:\:\:\:{with}\:\mathrm{0}<{x}<\mathrm{1} \\ $$

Question Number 62416 Answers: 0 Comments: 1

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>1 calculate Γ^((n)) (x) for all integr n.