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Question Number 160008 Answers: 2 Comments: 2

x_1 =3 ; n(x_1 +x_2 +...+x_n )=x_n ; n∈N ; n≥1 Find: Ω =Σ_(n=1) ^∞ (-1)^(n+1) x_n

$$\mathrm{x}_{\mathrm{1}} =\mathrm{3}\:;\:\mathrm{n}\left(\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +...+\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)=\mathrm{x}_{\boldsymbol{\mathrm{n}}} \:;\:\mathrm{n}\in\mathbb{N}\:;\:\mathrm{n}\geqslant\mathrm{1} \\ $$$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \\ $$

Question Number 160007 Answers: 0 Comments: 0

Find: lim_(n→∞) (n(((1 + (1/n))^n - e - 1)^n - e^(- (e/2)) ))

$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{n}\left(\left(\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{n}}\right)^{\boldsymbol{\mathrm{n}}} -\:\mathrm{e}\:-\:\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} -\:\mathrm{e}^{-\:\frac{\mathrm{e}}{\mathrm{2}}} \right)\right) \\ $$$$ \\ $$

Question Number 160006 Answers: 0 Comments: 2

Evaluate: lim_(n→∞) ∫_n ^(n+1) e^(1/x) dx = ?

$$\mathrm{Evaluate}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{n}}} {\overset{\boldsymbol{\mathrm{n}}+\mathrm{1}} {\int}}\:\mathrm{e}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \:\mathrm{dx}\:=\:? \\ $$$$ \\ $$

Question Number 159999 Answers: 1 Comments: 0

Question Number 159994 Answers: 1 Comments: 0

montrer que le quotient d′un nombe rationnel et dun nombre irrationnel est irrationnel

$$ \\ $$$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{le}\:\mathrm{quotient}\:\mathrm{d}'\mathrm{un} \\ $$$$\mathrm{nombe}\:\mathrm{rationnel}\:\mathrm{et}\:\mathrm{dun}\:\mathrm{nombre}\: \\ $$$$\mathrm{irr}{a}\mathrm{tionnel}\:\mathrm{est}\:\mathrm{irrationnel} \\ $$

Question Number 159973 Answers: 0 Comments: 6

Can anyone please resolve the Q 159787 in details..

$${Can}\:{anyone}\:{please}\:{resolve}\:{the} \\ $$$${Q}\:\mathrm{159787}\:{in}\:{details}.. \\ $$

Question Number 159966 Answers: 0 Comments: 2

a y=(√(x+(√(x+(√(x+.....)))))) b y=(√(x(√(x(√(x(√(x.....)))))))) find (dy/dx)

$${a}\:\:\:\:{y}=\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+.....}}} \\ $$$${b}\:\:\:\:\:{y}=\sqrt{{x}\sqrt{{x}\sqrt{{x}\sqrt{{x}.....}}}} \\ $$$${find}\:\frac{{dy}}{{dx}} \\ $$

Question Number 159962 Answers: 0 Comments: 0

show me: these are the cauchy criterion. please 1.(((n+1)/n)) 2. (1+(1/(2!))+(1/(3!))+...+(1/(n!))) 3. ((−1)^n ) 4. n+(((−1)^n )/n) 5.(1nm)

$${show}\:{me}:\:{these}\:{are}\:{the}\:{cauchy}\:{criterion}.\:{please} \\ $$$$\mathrm{1}.\left(\frac{{n}+\mathrm{1}}{{n}}\right) \\ $$$$\mathrm{2}.\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{3}!}+...+\frac{\mathrm{1}}{{n}!}\right) \\ $$$$\mathrm{3}.\:\left(\left(−\mathrm{1}\right)^{{n}} \right) \\ $$$$\mathrm{4}.\:{n}+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}} \\ $$$$\mathrm{5}.\left(\mathrm{1}{nm}\right) \\ $$$$ \\ $$

Question Number 159961 Answers: 4 Comments: 0

if x + (1/x) = 2(√5) then find the value of ((x(x^6 − 1))/(x^8 − 1))

$$\mathrm{if}\:{x}\:+\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{2}\sqrt{\mathrm{5}}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{{x}\left({x}^{\mathrm{6}} \:−\:\mathrm{1}\right)}{{x}^{\mathrm{8}} \:−\:\mathrm{1}} \\ $$

Question Number 159960 Answers: 1 Comments: 0

∫_0 ^( 1) ((3x^3 −x^2 +2x−4)/( (√(x^2 −3x+2)))) dx =?

$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{4}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}}\:{dx}\:=?\: \\ $$

Question Number 159958 Answers: 1 Comments: 3

find the area and perimeter of ((x/a))^(2/3) +((y/b))^(2/3) =1

$${find}\:{the}\:{area}\:{and}\:{perimeter}\:{of} \\ $$$$\left(\frac{\boldsymbol{{x}}}{\boldsymbol{{a}}}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\left(\frac{\boldsymbol{{y}}}{\boldsymbol{{b}}}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} =\mathrm{1} \\ $$

Question Number 159944 Answers: 0 Comments: 4

Question Number 159943 Answers: 1 Comments: 0

Question Number 159942 Answers: 0 Comments: 0

Question Number 159941 Answers: 1 Comments: 1

Question Number 159938 Answers: 1 Comments: 0

Evaluate ∫_1 ^( 4) (√((x−1)/x^5 )) dx.

$$\mathrm{Evaluate}\:\int_{\mathrm{1}} ^{\:\mathrm{4}} \sqrt{\frac{{x}−\mathrm{1}}{{x}^{\mathrm{5}} }}\:{dx}. \\ $$

Question Number 159936 Answers: 0 Comments: 0

Prove:: lim_(n→+∞) ^(−) nΣ_(k=n+1) ^n (((−1)^(k−1) )/k)=(1/2)

$$\mathrm{Prove}::\:\:\:\underset{\mathrm{n}\rightarrow+\infty} {\overline {\mathrm{lim}}n}\underset{\mathrm{k}=\mathrm{n}+\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} }{\mathrm{k}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 159935 Answers: 0 Comments: 0

Prove:: lim_(x→+∞) ^(−) xe^(−x) ∫_1 ^x ((e^t sin t)/t)dt=(1/( (√2)))

$$\mathrm{Prove}::\:\:\:\:\underset{\mathrm{x}\rightarrow+\infty} {\overline {\mathrm{lim}}xe}^{−\mathrm{x}} \int_{\mathrm{1}} ^{\mathrm{x}} \frac{\mathrm{e}^{\mathrm{t}} \mathrm{sin}\:\mathrm{t}}{\mathrm{t}}\mathrm{dt}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$

Question Number 159931 Answers: 0 Comments: 0

Prove :: ∫_0 ^∞ ((sin^n x)/x^m )dx=(1/(Γ(m)))∫_0 ^∞ ((D^(m−1) sin^n x)/x)dx n+m∈Odd Number.

$$\mathrm{Prove}\:::\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{\mathrm{n}} \mathrm{x}}{\mathrm{x}^{\mathrm{m}} }\mathrm{dx}=\frac{\mathrm{1}}{\Gamma\left(\mathrm{m}\right)}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{D}^{\mathrm{m}−\mathrm{1}} \mathrm{sin}^{\mathrm{n}} \mathrm{x}}{\mathrm{x}}\mathrm{dx} \\ $$$$\mathrm{n}+\mathrm{m}\in\mathrm{Odd}\:\mathrm{Number}. \\ $$

Question Number 159925 Answers: 1 Comments: 0

Find out pairs of numbers (a,b) (as many as you can) such that: (√a) +(√b) , a+b , a^2 +b^2 ∈ P

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{numbers}\:\left(\mathrm{a},\mathrm{b}\right)\:\left(\mathrm{as}\right. \\ $$$$\left.\mathrm{many}\:\mathrm{as}\:\mathrm{you}\:\mathrm{can}\right)\:\mathrm{such}\:\mathrm{that}: \\ $$$$\sqrt{\mathrm{a}}\:+\sqrt{\mathrm{b}}\:,\:\mathrm{a}+\mathrm{b}\:,\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \:\in\:\mathbb{P} \\ $$

Question Number 159921 Answers: 0 Comments: 0

Question Number 159918 Answers: 0 Comments: 2

y = sin 8x cos 4x y^((n)) =?

$$\:\:{y}\:=\:\mathrm{sin}\:\mathrm{8}{x}\:\mathrm{cos}\:\mathrm{4}{x}\: \\ $$$$\:\:{y}^{\left({n}\right)} \:=? \\ $$

Question Number 159917 Answers: 0 Comments: 1

Question Number 159915 Answers: 0 Comments: 1

∫_1 ^(16) ((√x)/(1+(x^3 )^(1/4) )) dx =?

$$\:\:\int_{\mathrm{1}} ^{\mathrm{16}} \:\frac{\sqrt{{x}}}{\mathrm{1}+\sqrt[{\mathrm{4}}]{{x}^{\mathrm{3}} }}\:{dx}\:=? \\ $$

Question Number 159928 Answers: 1 Comments: 0

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Question Number 159911 Answers: 1 Comments: 0

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