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Question Number 160844 Answers: 1 Comments: 0

Question Number 160823 Answers: 0 Comments: 1

Question Number 160689 Answers: 1 Comments: 0

Question Number 160672 Answers: 3 Comments: 1

Calculate lim_(x→0) ((tgx^m )/((sin x)^n )), lim_(x→0) ((xcos x−x)/((e^x −1)ln (1+3x^2 )))

$${Calculate}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{tgx}^{{m}} }{\left(\mathrm{sin}\:{x}\right)^{{n}} },\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\mathrm{cos}\:{x}−{x}}{\left({e}^{{x}} −\mathrm{1}\right)\mathrm{ln}\:\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} \right)} \\ $$

Question Number 160539 Answers: 1 Comments: 0

Prove by recurrence that (1/(n!))≤(1/2^(n−1) ), ∀n≥1.

$${Prove}\:{by}\:{recurrence}\:{that} \\ $$$$\frac{\mathrm{1}}{{n}!}\leqslant\frac{\mathrm{1}}{\mathrm{2}^{{n}−\mathrm{1}} },\:\forall{n}\geqslant\mathrm{1}. \\ $$

Question Number 160501 Answers: 1 Comments: 1

Calculate 1) lim_(x→1) ((cos ((Π/2))x)/(1−(√x))) 2) lim_(x→+∞) (e^(1+x) /((1+x)^x ))−(x/e)

$${Calculate} \\ $$$$\left.\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\mathrm{cos}\:\left(\frac{\Pi}{\mathrm{2}}\right){x}}{\mathrm{1}−\sqrt{{x}}} \\ $$$$\left.\mathrm{2}\right)\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\frac{{e}^{\mathrm{1}+{x}} }{\left(\mathrm{1}+{x}\right)^{{x}} }−\frac{{x}}{{e}} \\ $$

Question Number 160372 Answers: 1 Comments: 0

Calculate Σ_(k=0) ^(2000) i^k , Σ_(k=0) ^(2002) (−1)^k

$${Calculate} \\ $$$$\sum_{{k}=\mathrm{0}} ^{\mathrm{2000}} {i}^{{k}} ,\:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2002}} \left(−\mathrm{1}\right)^{{k}} \\ $$

Question Number 160361 Answers: 0 Comments: 0

1^o Prove by recrrence that , for n≥28 , n!≥11^n . 2^o Deduce the limite of the suite (((n!)/(10^n ))) when n tend verse +∞.

$$\mathrm{1}^{{o}} \:{Prove}\:{by}\:{recrrence}\:{that}\:,\:{for} \\ $$$${n}\geqslant\mathrm{28}\:,\:\:{n}!\geqslant\mathrm{11}^{{n}} . \\ $$$$\mathrm{2}^{{o}} \:{Deduce}\:{the}\:{limite}\:{of}\:{the}\:{suite} \\ $$$$\left(\frac{{n}!}{\mathrm{10}^{{n}} }\right)\:{when}\:{n}\:{tend}\:{verse}\:+\infty. \\ $$

Question Number 159881 Answers: 0 Comments: 0

Resolve 1. u_n −3u_(n−1) =12((3/4))^n 2. u_n =2u_(n−1) +5cos (((nΠ)/3)), u_o =1 3. u_n =u_(n−1) −u_(n−2) +2sin (((nΠ)/3)) with u_o =1, u_1 =2

$${Resolve}\: \\ $$$$\mathrm{1}.\:\:{u}_{{n}} −\mathrm{3}{u}_{{n}−\mathrm{1}} =\mathrm{12}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} \\ $$$$\mathrm{2}.\:{u}_{{n}} =\mathrm{2}{u}_{{n}−\mathrm{1}} +\mathrm{5cos}\:\left(\frac{{n}\Pi}{\mathrm{3}}\right),\:{u}_{{o}} =\mathrm{1}\: \\ $$$$\mathrm{3}.\:{u}_{{n}} ={u}_{{n}−\mathrm{1}} −{u}_{{n}−\mathrm{2}} +\mathrm{2sin}\:\left(\frac{{n}\Pi}{\mathrm{3}}\right) \\ $$$${with}\:{u}_{{o}} =\mathrm{1},\:{u}_{\mathrm{1}} =\mathrm{2} \\ $$

Question Number 159839 Answers: 1 Comments: 0

q=2(√(2(√2)))

$${q}=\mathrm{2}\sqrt{\mathrm{2}\sqrt{\mathrm{2}}} \\ $$

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 159675 Answers: 1 Comments: 2

Question Number 159560 Answers: 1 Comments: 0

Resolve 1. u_(n+2) −2u_(n+1) +4u_n =3^n with u_o =1, u_1 =−2 2. u_n =u_(n−1) −u_(n−2) +2sin (((nΠ)/3)) with u_o =1, u_1 =2

$${Resolve}\: \\ $$$$\mathrm{1}.\:{u}_{{n}+\mathrm{2}} −\mathrm{2}{u}_{{n}+\mathrm{1}} +\mathrm{4}{u}_{{n}} =\mathrm{3}^{{n}} \\ $$$${with}\:{u}_{{o}} =\mathrm{1},\:{u}_{\mathrm{1}} =−\mathrm{2} \\ $$$$\mathrm{2}.\:{u}_{{n}} ={u}_{{n}−\mathrm{1}} −{u}_{{n}−\mathrm{2}} +\mathrm{2sin}\:\left(\frac{{n}\Pi}{\mathrm{3}}\right) \\ $$$${with}\:{u}_{{o}} =\mathrm{1},\:{u}_{\mathrm{1}} =\mathrm{2} \\ $$

Question Number 159475 Answers: 0 Comments: 0

Question Number 159425 Answers: 0 Comments: 0

Question Number 159403 Answers: 0 Comments: 1

U_(n+1) =(1/2)(u_n +(a/u_n )) with u_1 >0, a>0 Prove that (u_(n+1) /u_n )≤1

$${U}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}} +\frac{{a}}{{u}_{{n}} }\right)\:{with}\:{u}_{\mathrm{1}} >\mathrm{0},\:\:{a}>\mathrm{0} \\ $$$${Prove}\:{that}\:\:\frac{{u}_{{n}+\mathrm{1}} }{{u}_{{n}} }\leqslant\mathrm{1} \\ $$

Question Number 159309 Answers: 1 Comments: 0

Resolve I_n =∫_(−1) ^1 (1−x^2 )^n dx

$${Resolve}\:{I}_{{n}} =\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx} \\ $$

Question Number 159171 Answers: 1 Comments: 0

Prove by absurd that log 2 is the number irrational

$${Prove}\:{by}\:{absurd}\:{that}\:\mathrm{log}\:\mathrm{2}\:{is}\:{the} \\ $$$${number}\:{irrational} \\ $$

Question Number 159123 Answers: 2 Comments: 0

Question Number 159078 Answers: 0 Comments: 0

1) Prove by recurrence that for n≥28, n!≥11^n 2) On subtract the limit of the suite (((n!)/(10^n ))) when n tended at +∞

$$\left.\mathrm{1}\right)\:{Prove}\:{by}\:{recurrence}\:{that}\: \\ $$$${for}\:{n}\geqslant\mathrm{28},\:\:\:{n}!\geqslant\mathrm{11}^{{n}} \: \\ $$$$\left.\mathrm{2}\right)\:{On}\:{subtract}\:{the}\:{limit}\:{of}\:{the}\: \\ $$$${suite}\:\left(\frac{{n}!}{\mathrm{10}^{{n}} }\right)\:{when}\:{n}\:{tended}\:{at}\:+\infty \\ $$

Question Number 159016 Answers: 0 Comments: 0

Question Number 158984 Answers: 0 Comments: 0

1. Prove by recurrence that so n ∈ N and θ ∈ R (cos (nθ)+isin (nθ)=cos (nθ)+isin (nθ) 2. Prove that U_(n+1) =(1/5)(U_n ^2 +6) and U_1 =(5/2), is decrease

$$\mathrm{1}.\:{Prove}\:{by}\:{recurrence}\:{that}\: \\ $$$${so}\:\:{n}\:\in\:{N}\:{and}\:\theta\:\in\: {R}\: \\ $$$$\left(\mathrm{cos}\:\left({n}\theta\right)+{i}\mathrm{sin}\:\left({n}\theta\right)=\mathrm{cos}\:\left({n}\theta\right)+{i}\mathrm{sin}\:\left({n}\theta\right)\right. \\ $$$$\mathrm{2}.\:{Prove}\:{that}\:{U}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{5}}\left({U}_{{n}} ^{\mathrm{2}} +\mathrm{6}\right)\:{and} \\ $$$${U}_{\mathrm{1}} =\frac{\mathrm{5}}{\mathrm{2}},\:{is}\:{decrease} \\ $$$$ \\ $$

Question Number 158945 Answers: 2 Comments: 0

1) Prove by absurd that ((ln 2)/(ln 3)) is irrational 2) Prove by absurd that (√2)+(√(6 ))≤(√(15))

$$\left.\mathrm{1}\right)\:{Prove}\:{by}\:{absurd}\:{that}\: \\ $$$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:\mathrm{3}}\:\:{is}\:{irrational} \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{by}\:{absurd}\:{that} \\ $$$$\sqrt{\mathrm{2}}+\sqrt{\mathrm{6}\:}\leqslant\sqrt{\mathrm{15}} \\ $$

Question Number 158858 Answers: 0 Comments: 0

I_n =∫_(−1) ^1 (1−x^2 )^n cos ((a/(2b))x)dx to integrating by piece for n≥2 proven (a^2 /(4b^2 ))I_(n ) =2n(2n−1)I_(n−1) −4(n−1)I_(n−2) proven by rearring that ((a/(2b)))^(2n+1) I_n =n![p((q/(2b)))sin ((a/(2b)))+Q((a/(2b)))cos ((a/(2b)))]

$${I}_{{n}} =\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} \mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}{x}\right){dx} \\ $$$${to}\:{integrating}\:{by}\:{piece}\:{for}\:{n}\geqslant\mathrm{2}\: \\ $$$${proven}\: \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{4}{b}^{\mathrm{2}} }{I}_{{n}\:} =\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right){I}_{{n}−\mathrm{1}} −\mathrm{4}\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} \\ $$$${proven}\:{by}\:{rearring}\:{that}\: \\ $$$$\left(\frac{{a}}{\mathrm{2}{b}}\right)^{\mathrm{2}{n}+\mathrm{1}} {I}_{{n}} ={n}!\left[{p}\left(\frac{{q}}{\mathrm{2}{b}}\right)\mathrm{sin}\:\left(\frac{{a}}{\mathrm{2}{b}}\right)+{Q}\left(\frac{{a}}{\mathrm{2}{b}}\right)\mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}\right)\right] \\ $$

Question Number 158855 Answers: 1 Comments: 0

Resolve the system d′ unknow (x, y,z) ∈ ⊂^3 x+y+z=1 x^2 +y^2 +z^2 =1 x^3 +y^3 +z^3 =−5

$${Resolve}\:{the}\:{system}\:{d}'\:{unknow}\:\:\left({x},\:{y},{z}\right)\:\in\:\subset^{\mathrm{3}} \\ $$$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{1} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =−\mathrm{5} \\ $$

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