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Question Number 74799 Answers: 1 Comments: 0

prove that 0≤∫_0 ^∞ ((t^2 e^(−nt) )/(e^t −1))dt ≤(1/n^2 ) for n integr not 0

$${prove}\:{that}\:\mathrm{0}\leqslant\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\mathrm{2}} \:{e}^{−{nt}} }{{e}^{{t}} −\mathrm{1}}{dt}\:\leqslant\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:\:{for}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$

Question Number 74798 Answers: 0 Comments: 0

calculate ∫_0 ^π ln(1−2xcosθ +x^2 )dθ with ∣x∣<1

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

Question Number 74797 Answers: 0 Comments: 0

find nsture of the serie Σ sin(πen!)

$${find}\:{nsture}\:{of}\:{the}\:{serie}\:\Sigma\:{sin}\left(\pi{en}!\right) \\ $$

Question Number 74796 Answers: 0 Comments: 0

let U_n =(−1)^n {arcsin((1/n))−(1/n)}^(1/3) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \left\{{arcsin}\left(\frac{\mathrm{1}}{{n}}\right)−\frac{\mathrm{1}}{{n}}\right\}^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 74795 Answers: 1 Comments: 1

study the convergence of Σ (1/(nH_n )) with H_n =Σ_(k=1) ^n (1/k)

$${study}\:{the}\:{convergence}\:{of}\:\Sigma\:\frac{\mathrm{1}}{{nH}_{{n}} } \\ $$$${with}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 74794 Answers: 0 Comments: 2

let A = (((1 2)),((−1 1)) ) 1) calculate A^n 2) find e^A and e^(−A) 3) calculate cosA and sinA 4) calculate ch(A) and sh(A)

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{A}} \:{and}\:{e}^{−{A}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:{cosA}\:{and}\:{sinA} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:{ch}\left({A}\right)\:{and}\:{sh}\left({A}\right) \\ $$

Question Number 74790 Answers: 0 Comments: 0

4x^4 +12x3+25x2t+24x+16 find the square root

$$\mathrm{4}{x}^{\mathrm{4}} +\mathrm{12}{x}\mathrm{3}+\mathrm{25}{x}\mathrm{2}{t}+\mathrm{24}{x}+\mathrm{16}\:{find}\:{the}\:{square}\:{root} \\ $$

Question Number 74786 Answers: 2 Comments: 1

Question Number 74793 Answers: 1 Comments: 1

prove the convergence of ∫_0 ^1 ((ln(1+(√x)))/(√x))dx

$${prove}\:{the}\:{convergence}\:{of}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\sqrt{{x}}\right)}{\sqrt{{x}}}{dx} \\ $$

Question Number 74782 Answers: 1 Comments: 2

Question Number 74776 Answers: 1 Comments: 0

Question Number 74774 Answers: 0 Comments: 0

Question Number 74766 Answers: 0 Comments: 2

Question Number 74753 Answers: 1 Comments: 4

evaluate 5^(√(log 7_5 )) − 7^(√(log 5_7 ))

$${evaluate}\:\mathrm{5}^{\sqrt{\mathrm{log}\:\mathrm{7}_{\mathrm{5}} }} \:−\:\mathrm{7}^{\sqrt{\mathrm{log}\:\mathrm{5}_{\mathrm{7}} }} \\ $$

Question Number 74748 Answers: 1 Comments: 3

Question Number 74747 Answers: 1 Comments: 0

Question Number 74742 Answers: 2 Comments: 1

If α and β are the roots of x^2 − x + 1 = 0, Find α^(23) + β^(23) without demoivre′s theorem.

$$\mathrm{If}\:\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{x}\:+\:\mathrm{1}\:\:\:=\:\:\mathrm{0},\:\: \\ $$$$\mathrm{Find}\:\:\:\:\:\:\:\:\:\alpha^{\mathrm{23}} \:+\:\beta^{\mathrm{23}} \:\:\:\:\:\mathrm{without}\:\mathrm{demoivre}'\mathrm{s}\:\mathrm{theorem}. \\ $$

Question Number 74723 Answers: 1 Comments: 1

3xy+x^2 +y^2 =5 find the second derivative

$$\mathrm{3}{xy}+{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{5} \\ $$$${find}\:{the}\:{second}\:{derivative} \\ $$

Question Number 74720 Answers: 2 Comments: 5

Question Number 74716 Answers: 1 Comments: 0

Question Number 74713 Answers: 1 Comments: 1

Question Number 74712 Answers: 0 Comments: 2

Question Number 74711 Answers: 0 Comments: 0

Question Number 74944 Answers: 0 Comments: 0

∫(e^(−cos(2x)) /(sin^2 (x))) dx

$$\int\frac{{e}^{−{cos}\left(\mathrm{2}{x}\right)} }{{sin}^{\mathrm{2}} \left({x}\right)}\:{dx} \\ $$

Question Number 74726 Answers: 1 Comments: 3

Question Number 74703 Answers: 1 Comments: 0

let b and r be two positive prime numbers such that b≠r and b×r is a divisor of 138. Consider an arithmetic progression in which the first term is b, the ratio is r and the fourth term is 71. What is the value of b+r?

$${let}\:\boldsymbol{{b}}\:{and}\:\boldsymbol{{r}}\:{be}\:{two}\:{positive}\:{prime}\: \\ $$$${numbers}\:{such}\:{that}\:{b}\neq{r}\:{and}\:{b}×{r}\:{is} \\ $$$${a}\:{divisor}\:{of}\:\mathrm{138}.\:{Consider}\:{an}\: \\ $$$${arithmetic}\:{progression}\:{in}\:{which} \\ $$$${the}\:{first}\:{term}\:{is}\:\boldsymbol{{b}},\:{the}\:{ratio}\:{is}\:\boldsymbol{{r}} \\ $$$${and}\:{the}\:{fourth}\:{term}\:{is}\:\mathrm{71}.\:{What}\:{is}\:{the} \\ $$$${value}\:{of}\:\boldsymbol{{b}}+\boldsymbol{{r}}? \\ $$

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