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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.

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} {e}^{−{t}} \:{dt}\:\:\:{with}\:{x}>\mathrm{1}\:{calculate}\:\Gamma^{\left({n}\right)} \left({x}\right)\:{for}\:{all}\:{integr}\:{n}. \\ $$

Question Number 62415 Answers: 0 Comments: 1

calculate f(x,y) =∫_0 ^∞ e^(−xt) ln(yt) dt with x>0 and y>0 .

$${calculate}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{e}^{−{xt}} {ln}\left({yt}\right)\:{dt}\:\:{with}\:{x}>\mathrm{0}\:{and}\:{y}>\mathrm{0}\:. \\ $$

Question Number 62414 Answers: 1 Comments: 0

find ∫ (e^x /(√(e^(2x) −1)))dx

$${find}\:\int\:\:\:\:\:\frac{{e}^{{x}} }{\sqrt{{e}^{\mathrm{2}{x}} −\mathrm{1}}}{dx} \\ $$

Question Number 62413 Answers: 0 Comments: 0

calculate W_n = ∫_0 ^(π/2) cos^n xdx ( n from N) and J_n =∫_0 ^(π/2) sin^n xdx

$${calculate}\:\:{W}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{{n}} {xdx}\:\:\:\left(\:{n}\:{from}\:{N}\right)\:{and}\:{J}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}^{{n}} {xdx} \\ $$

Question Number 62412 Answers: 0 Comments: 2

calculate lim_(n→+∞) ∫_0 ^n (1−(x/n))^n dx

$${calculate}\:\:{lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {dx} \\ $$

Question Number 62410 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−t) ln(t) dt =−γ ( γ is the constant of euler)

$$\:\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}} {ln}\left({t}\right)\:{dt}\:=−\gamma\:\:\:\:\:\:\:\left(\:\:\gamma\:{is}\:{the}\:{constant}\:{of}\:{euler}\right) \\ $$

Question Number 62399 Answers: 1 Comments: 0

What the definition of Claim , Theorem , and Lemma ? When can we use them respectively for getting proof(s) ?

$${What}\:\:{the}\:\:{definition}\:\:{of}\:\:{Claim}\:,\:{Theorem}\:,\:\:{and}\:\:{Lemma}\:\:? \\ $$$${When}\:\:{can}\:\:{we}\:\:{use}\:\:{them}\:\:{respectively}\:\:{for}\:\:{getting}\:\:{proof}\left({s}\right)\:? \\ $$

Question Number 62396 Answers: 0 Comments: 0

Question Number 62395 Answers: 0 Comments: 1

The Most Beautiful Equation for me is: e^(iπ) +1=0 INCREDIBLE! #Euler′sIdentity

$$\mathrm{The}\:\mathrm{Most}\:\mathrm{Beautiful}\:\mathrm{Equation} \\ $$$$\mathrm{for}\:\mathrm{me}\:\mathrm{is}: \\ $$$$\mathrm{e}^{{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{INCREDIBLE}! \\ $$$$#\mathrm{Euler}'\mathrm{sIdentity} \\ $$

Question Number 62389 Answers: 1 Comments: 1

∫0dx= help

$$\int\mathrm{0dx}= \\ $$$$ \\ $$$$ \\ $$$$\mathrm{help} \\ $$

Question Number 62388 Answers: 1 Comments: 1

If tan θ=(1/2) and tan φ=(1/3), then the value of θ + φ is

$$\mathrm{If}\:\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{tan}\:\phi=\frac{\mathrm{1}}{\mathrm{3}},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\theta\:+\:\phi\:\:\:\mathrm{is} \\ $$

Question Number 62380 Answers: 1 Comments: 0

Prove that if the lengths of a triangle form an arithmetic progression, then the centre of incircle and the centroid of triangle lie on a line parallel to the side of middle length of the triangle.

$${Prove}\:{that}\:{if}\:{the}\:{lengths}\:{of}\:{a}\: \\ $$$${triangle}\:{form}\:{an}\:{arithmetic} \\ $$$${progression},\:{then}\:{the}\:{centre}\:{of} \\ $$$${incircle}\:{and}\:{the}\:{centroid}\:{of} \\ $$$${triangle}\:{lie}\:{on}\:{a}\:{line}\:{parallel}\:{to} \\ $$$${the}\:{side}\:{of}\:{middle}\:{length}\:{of}\:{the} \\ $$$${triangle}. \\ $$

Question Number 62372 Answers: 1 Comments: 0

Solve for x , y 3x>2y ∧ 2x<3y where x,y∈N

$${Solve}\:{for}\:{x}\:,\:{y} \\ $$$$\mathrm{3}{x}>\mathrm{2}{y}\:\wedge\:\mathrm{2}{x}<\mathrm{3}{y}\: \\ $$$${where}\:{x},{y}\in\mathbb{N} \\ $$

Question Number 62363 Answers: 2 Comments: 0

Question Number 62347 Answers: 2 Comments: 0

Question Number 62343 Answers: 1 Comments: 1

calculate ∫_0 ^(π/4) {xΠ_(k=1) ^∞ cos((x/2^k ))}dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left\{{x}\prod_{{k}=\mathrm{1}} ^{\infty} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\right\}{dx} \\ $$

Question Number 62342 Answers: 1 Comments: 4

let f(ξ) =∫ (x^2 /(√(1−ξx^2 )))dx with 0<ξ<1 1) determine a explicit form of f(ξ) 2) calculate lim_(ξ→1) f(ξ) 3) calculate ∫_0 ^(1/2) (x^2 /(√(1−sin^2 θ x^2 ))) dx with 0<θ<(π/2)

$${let}\:{f}\left(\xi\right)\:=\int\:\:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}{dx}\:\:\:{with}\:\:\mathrm{0}<\xi<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{\xi\rightarrow\mathrm{1}} \:\:\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−{sin}^{\mathrm{2}} \theta\:{x}^{\mathrm{2}} }}\:{dx}\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

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