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Question Number 159775    Answers: 1   Comments: 0

Π_(n=1) ^∞ ((α^3 +β^2 )/3^n )= ? in expanded form

$$\prod_{\mathrm{n}=\mathrm{1}} ^{\infty} \frac{\alpha^{\mathrm{3}} +\beta^{\mathrm{2}} }{\mathrm{3}^{\mathrm{n}} }=\:? \\ $$$$\mathrm{in}\:\mathrm{expanded}\:\mathrm{form} \\ $$

Question Number 159781    Answers: 0   Comments: 0

Question Number 159771    Answers: 0   Comments: 1

Question Number 159768    Answers: 1   Comments: 1

Question Number 159763    Answers: 0   Comments: 4

Question Number 159762    Answers: 1   Comments: 0

Given log _3 (n)= log _6 (m)=log _(12) (m+n) (m/n) = ?

$$\:\:\:{Given}\:\mathrm{log}\:_{\mathrm{3}} \left({n}\right)=\:\mathrm{log}\:_{\mathrm{6}} \left({m}\right)=\mathrm{log}\:_{\mathrm{12}} \left({m}+{n}\right) \\ $$$$\:\:\:\frac{{m}}{{n}}\:=\:? \\ $$

Question Number 159759    Answers: 1   Comments: 0

Question Number 159757    Answers: 0   Comments: 1

Question Number 159755    Answers: 1   Comments: 1

Question Number 159754    Answers: 1   Comments: 0

lim_(x→0^+ ) (((√(tan x)) + (√(sin x)) −2(√x))/( (√(sin x)) − (√(tan x)) )) = ?

$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\sqrt{\mathrm{tan}\:{x}}\:+\:\sqrt{\mathrm{sin}\:{x}}\:−\mathrm{2}\sqrt{{x}}}{\:\sqrt{\mathrm{sin}\:{x}}\:−\:\sqrt{\mathrm{tan}\:{x}}\:}\:=\:? \\ $$

Question Number 159750    Answers: 0   Comments: 1

Prove that (√(689×690×691+1)) is a naturel number.

$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\:\sqrt{\mathrm{689}×\mathrm{690}×\mathrm{691}+\mathrm{1}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{naturel}}\:\boldsymbol{\mathrm{number}}. \\ $$

Question Number 159748    Answers: 2   Comments: 0

Question Number 159747    Answers: 0   Comments: 0

Question Number 159744    Answers: 1   Comments: 0

Question Number 159742    Answers: 1   Comments: 0

Find the perimeter of the figure which is bounded with the curves y^3 =x^2 and y=(√(2−x^2 ))

$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{perimeter}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{figure}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{is}} \\ $$$$\boldsymbol{\mathrm{bounded}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{curves}}\:\boldsymbol{\mathrm{y}}^{\mathrm{3}} =\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}=\sqrt{\mathrm{2}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \\ $$

Question Number 159740    Answers: 0   Comments: 0

Question Number 159737    Answers: 0   Comments: 0

Prove 1) E(x)+E(y)≤E(x+y)≤E(x)+E(y)+1 2) E(x)+E(y)+E(x+1)≤E(2x)+E(2y) 3) E((x/2))+E(((x+1)/2))=E(x)

$${Prove}\: \\ $$$$\left.\mathrm{1}\right)\:{E}\left({x}\right)+{E}\left({y}\right)\leqslant{E}\left({x}+{y}\right)\leqslant{E}\left({x}\right)+{E}\left({y}\right)+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{E}\left({x}\right)+{E}\left({y}\right)+{E}\left({x}+\mathrm{1}\right)\leqslant{E}\left(\mathrm{2}{x}\right)+{E}\left(\mathrm{2}{y}\right) \\ $$$$\left.\mathrm{3}\right)\:{E}\left(\frac{{x}}{\mathrm{2}}\right)+{E}\left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)={E}\left({x}\right) \\ $$

Question Number 159736    Answers: 0   Comments: 0

Prove that 1)Sup(A∪B)=max(Sup(A), Sup(B)) 2) inf(A∪B)=min(inf(A), inf(B))

$${Prove}\:{that} \\ $$$$\left.\mathrm{1}\right){Sup}\left({A}\cup{B}\right)={ma}\mathrm{x}\left(\mathrm{S}{up}\left({A}\right),\:{Sup}\left({B}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{inf}\left({A}\cup{B}\right)={min}\left({inf}\left({A}\right),\:{inf}\left({B}\right)\right) \\ $$

Question Number 159733    Answers: 0   Comments: 0

lim_(x→0^+ ) ((⌊xcos(x)⌋)/(xsin(π⌊(e^(1/x) /(ln(x)))⌋)))

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\lfloor{xcos}\left({x}\right)\rfloor}{{xsin}\left(\pi\lfloor\frac{{e}^{\frac{\mathrm{1}}{{x}}} }{{ln}\left({x}\right)}\rfloor\right)} \\ $$

Question Number 159731    Answers: 0   Comments: 0

Question Number 159730    Answers: 0   Comments: 0

Question Number 159727    Answers: 1   Comments: 0

lim_(x→0) ((1+tan (1−((x/(sin x))))))^(1/x^3 ) ?

$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{{x}^{\mathrm{3}} }]{\mathrm{1}+\mathrm{tan}\:\left(\mathrm{1}−\left(\frac{{x}}{\mathrm{sin}\:{x}}\right)\right)}\:?\: \\ $$

Question Number 159725    Answers: 0   Comments: 0

Question Number 159724    Answers: 0   Comments: 0

Question Number 159723    Answers: 1   Comments: 1

(1/(k+1))≤∫_k ^(k+1) ((1/x))dx≤(1/k) please show it with k∈ℵ−(0)

$$\frac{\mathrm{1}}{{k}+\mathrm{1}}\leqslant\int_{{k}} ^{{k}+\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right){dx}\leqslant\frac{\mathrm{1}}{{k}}\:\:\:\:\:\:\:\:\: \\ $$$$\:{please}\:{show}\:{it}\:{with}\:{k}\in\aleph−\left(\mathrm{0}\right) \\ $$$$ \\ $$

Question Number 159720    Answers: 1   Comments: 1

L = lim_(x→(π/3)) ((3−4sin^2 x)/(sin 2x−sin x)) ? Q = lim_(x→0) [(1/x^2 ) ((2/(cos^2 x)) +cos x−3)] ?

$$\:\:\:\:\:\:\:\:{L}\:=\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {\mathrm{lim}}\:\frac{\mathrm{3}−\mathrm{4sin}\:^{\mathrm{2}} {x}}{\mathrm{sin}\:\mathrm{2}{x}−\mathrm{sin}\:{x}}\:? \\ $$$$\:\:\:\:\:\:{Q}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:\left(\frac{\mathrm{2}}{\mathrm{cos}\:^{\mathrm{2}} {x}}\:+\mathrm{cos}\:{x}−\mathrm{3}\right)\right]\:?\: \\ $$

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