Question and Answers Forum

AllQuestion and Answers: Page 1398

Question Number 73334 Answers: 1 Comments: 0

calculate lim_(n→+∞) n^2 ( e^(sin((π/n^2 ))) −cos((π/n)))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:{n}^{\mathrm{2}} \left(\:{e}^{{sin}\left(\frac{\pi}{{n}^{\mathrm{2}} }\right)} −{cos}\left(\frac{\pi}{{n}}\right)\right) \\ $$

Question Number 73333 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(π +2x^2 ))/((x^2 +4)^2 ))dx

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

Question Number 73332 Answers: 0 Comments: 0

let U_n =Σ_(k=0) ^n (((−1)^k )/(√(2k+1))) determine a equivalent of n when n→+∞

$${let}\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{\mathrm{2}{k}+\mathrm{1}}}\:\:{determine}\:{a}\:{equivalent}\:{of}\:{n}\:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 73331 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((ln(1+e^(−3x^2 ) ))/(3+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{x}^{\mathrm{2}} } \right)}{\mathrm{3}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 73330 Answers: 1 Comments: 1

find lim_(x→0) ((ln(2−cos(2x)))/(ln(1+xsin(3x))))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left(\mathrm{2}−{cos}\left(\mathrm{2}{x}\right)\right)}{{ln}\left(\mathrm{1}+{xsin}\left(\mathrm{3}{x}\right)\right)} \\ $$

Question Number 73327 Answers: 0 Comments: 1

let w(x)=∫_0 ^∞ ((lnt)/((x^2 +t^2 )^2 ))dt 1) explicit w(x) 2) calculate U_n =∫_0 ^∞ ((lnt)/((n^2 +t^2 )^2 ))dt find lim_(n→+∞) n^4 U_n and determine nature of tbe serie Σ U_n

$${let}\:{w}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{explicit}\:{w}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left({n}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{4}} {U}_{{n}} \:\:{and}\:{determine}\:{nature}\:{of}\:{tbe}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 73308 Answers: 0 Comments: 6

what are the solutions of (√(3x^2 +1))=n where n∈N

$${what}\:{are}\:{the}\:{solutions} \\ $$$${of}\:\sqrt{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{1}}={n}\:{where}\:{n}\in\mathbb{N} \\ $$

Question Number 73297 Answers: 2 Comments: 1

Question Number 73295 Answers: 1 Comments: 0

How many solution so that 3n−4, 4n−5, 5n−13 are prime numbers ?

$${How}\:\:{many}\:\:{solution}\:\:{so}\:\:{that} \\ $$$$\mathrm{3}{n}−\mathrm{4},\:\:\mathrm{4}{n}−\mathrm{5},\:\:\mathrm{5}{n}−\mathrm{13} \\ $$$${are}\:\:{prime}\:\:{numbers}\:? \\ $$

Question Number 73293 Answers: 2 Comments: 2

Explicit f(x)= ∫_1 ^∞ ((lnt)/((x^2 +t^2 )^2 )) dt

$${Explicit}\:\:{f}\left({x}\right)=\:\int_{\mathrm{1}} ^{\infty} \:\frac{{lnt}}{\left({x}^{\mathrm{2}} +{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\: \\ $$

Question Number 73279 Answers: 1 Comments: 0

Question Number 73275 Answers: 0 Comments: 2

∫(4/(x^2 (√(4−xδϰ)))) ?

$$ \\ $$$$ \\ $$$$\int\frac{\mathrm{4}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}−{x}\delta\varkappa}}\:\:\:\:? \\ $$$$ \\ $$

Question Number 73274 Answers: 0 Comments: 2

Question Number 73273 Answers: 1 Comments: 0

Question Number 73269 Answers: 0 Comments: 0

Question Number 73260 Answers: 1 Comments: 0

Question Number 73255 Answers: 1 Comments: 0

Question Number 73247 Answers: 1 Comments: 0

Question Number 73243 Answers: 1 Comments: 0

prove that for z ∈C arctanz =(1/(2i))ln(((1+iz)/(1−iz)))

$${prove}\:{that}\:\:{for}\:{z}\:\in{C}\:\:\:{arctanz}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}{ln}\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right) \\ $$$$ \\ $$

Question Number 73238 Answers: 1 Comments: 1

let 0<a<1 calculate ∫_0 ^∞ ((ln(t)t^(a−1) )/(1+t))dt and ∫_0 ^∞ ((ln^2 (t)t^(a−1) )/(1+t))dt

$${let}\:\mathrm{0}<{a}<\mathrm{1}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}^{\mathrm{2}} \left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt} \\ $$

Question Number 73235 Answers: 0 Comments: 0

Question Number 73261 Answers: 1 Comments: 1

calculate Σ_(n=1) ^∞ (n^4 +2n^2 −3)(x^n /(n!)) in case of convergence.

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left({n}^{\mathrm{4}} +\mathrm{2}{n}^{\mathrm{2}} −\mathrm{3}\right)\frac{{x}^{{n}} }{{n}!}\:{in}\:{case}\:{of}\:{convergence}. \\ $$

Question Number 73231 Answers: 1 Comments: 1

find the sum of Σ_(n=0) ^∞ (n^2 −3n+1)e^(−n)

$${find}\:{the}\:{sum}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \left({n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{1}\right){e}^{−{n}} \\ $$

Question Number 73230 Answers: 0 Comments: 0

calculate A_n =∫_0 ^∞ ((1+x^n )/(2+x^(2n) ))dx and J_n =∫_0 ^∞ ((2+x^(3n) )/(5+x^(7n) ))dx with n integr natural not 0

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}^{{n}} }{\mathrm{2}+{x}^{\mathrm{2}{n}} }{dx}\:\:{and}\:{J}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}+{x}^{\mathrm{3}{n}} }{\mathrm{5}+{x}^{\mathrm{7}{n}} }{dx} \\ $$$${with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0} \\ $$

Question Number 73227 Answers: 1 Comments: 0

find the polynom T_n wich verify T_n (cosθ)=cos(nθ) ∀n integr ∀θ real 1) find T_0 ,T_1 and T_2 and prove that T_(n+2) =2x T_(n+1) −T_n 3) find deg(T_n ) and T_n (1) ,T_n (−1) 4) find T^′ (cosθ) for 0<θ<π and prove that (1−x^2 )T_n ′′−xT′_n +n^2 T_n =0 5) find roots of T_n and decompose T_n inside R[x] 6) find the value of Π_(k=0) ^(n−1) cos((((2k+1)π)/(2n)))

$${find}\:{the}\:{polynom}\:{T}_{{n}} \:{wich}\:{verify}\:{T}_{{n}} \left({cos}\theta\right)={cos}\left({n}\theta\right) \\ $$$$\forall{n}\:{integr}\:\forall\theta\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{T}_{\mathrm{0}} ,{T}_{\mathrm{1}} {and}\:{T}_{\mathrm{2}} {and}\:{prove}\:{that}\: \\ $$$${T}_{{n}+\mathrm{2}} =\mathrm{2}{x}\:{T}_{{n}+\mathrm{1}} −{T}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{deg}\left({T}_{{n}} \right)\:{and}\:{T}_{{n}} \left(\mathrm{1}\right)\:,{T}_{{n}} \left(−\mathrm{1}\right) \\ $$$$\left.\mathrm{4}\right)\:{find}\:{T}\:^{'} \left({cos}\theta\right)\:{for}\:\mathrm{0}<\theta<\pi\:\:{and}\:{prove}\:{that} \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){T}_{{n}} ''−{xT}'_{{n}} \:+{n}^{\mathrm{2}} \:{T}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{roots}\:{of}\:{T}_{{n}} \:{and}\:{decompose}\:{T}_{{n}} {inside}\:{R}\left[{x}\right] \\ $$$$\left.\mathrm{6}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\left(\frac{\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{\mathrm{2}{n}}\right) \\ $$

Question Number 73225 Answers: 0 Comments: 0

calculate f(x)=∫_0 ^π ln(x^2 −2xcosθ +1)dθ with x real.

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:{x}\:{real}. \\ $$

Pg 1393 Pg 1394 Pg 1395 Pg 1396 Pg 1397 Pg 1398 Pg 1399 Pg 1400 Pg 1401 Pg 1402

Contact: info@tinkutara.com