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Question Number 80828    Answers: 0   Comments: 2

given lim_(x→−∞) [(√(2x+p)) −(√(2x+1))]×(√(2x−p ))= (1/4)p find p .

$${given}\: \\ $$$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\left[\sqrt{\mathrm{2}{x}+{p}}\:−\sqrt{\mathrm{2}{x}+\mathrm{1}}\right]×\sqrt{\mathrm{2}{x}−{p}\:}=\:\frac{\mathrm{1}}{\mathrm{4}}{p} \\ $$$${find}\:{p}\:. \\ $$

Question Number 80823    Answers: 1   Comments: 0

show that lim_(x→∞) H_n =2F_1 (1,1;2,1) ln(4)−2ln(3)=2F_1 (1,1;2;((−1)/2))

$${show}\:{that}\: \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\:{H}_{{n}} =\mathrm{2}{F}_{\mathrm{1}} \left(\mathrm{1},\mathrm{1};\mathrm{2},\mathrm{1}\right) \\ $$$$ \\ $$$${ln}\left(\mathrm{4}\right)−\mathrm{2}{ln}\left(\mathrm{3}\right)=\mathrm{2}{F}_{\mathrm{1}} \left(\mathrm{1},\mathrm{1};\mathrm{2};\frac{−\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 80816    Answers: 1   Comments: 2

let 0<a<b prove that ln(1+(a/b))ln(1+(b/a))< (ln2)^2

$${let}\:\:\mathrm{0}<{a}<{b}\:\:{prove}\:{that} \\ $$$$\:{ln}\left(\mathrm{1}+\frac{{a}}{{b}}\right){ln}\left(\mathrm{1}+\frac{{b}}{{a}}\right)<\:\left({ln}\mathrm{2}\right)^{\mathrm{2}} \:\: \\ $$

Question Number 80815    Answers: 0   Comments: 0

find nsture of the serie Σ_(n=0) ^∞ (1+sin(n))^(1/n)

$${find}\:{nsture}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(\mathrm{1}+{sin}\left({n}\right)\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 80814    Answers: 0   Comments: 2

find nature of the serie Σ_(n=1) ^∞ (1−cos((π/n)))

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−{cos}\left(\frac{\pi}{{n}}\right)\right) \\ $$

Question Number 80813    Answers: 1   Comments: 0

calculate Σ_(n=2) ^∞ ((ξ(n)−1)/n) with ξ(x)=Σ_(n=1) ^∞ (1/n^x ) (x>1)

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\xi\left({n}\right)−\mathrm{1}}{{n}}\:\:\:{with}\:\xi\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\left({x}>\mathrm{1}\right) \\ $$

Question Number 80809    Answers: 1   Comments: 5

Question Number 80804    Answers: 2   Comments: 2

Find [(√1)]+[(√2)]+[(√3)]+...+[(√(100))]=? with [x]=greatest integer function can we find a general formula for [(√1)]+[(√2)]+[(√3)]+...+[(√n)] in terms of n?

$${Find} \\ $$$$\left[\sqrt{\mathrm{1}}\right]+\left[\sqrt{\mathrm{2}}\right]+\left[\sqrt{\mathrm{3}}\right]+...+\left[\sqrt{\mathrm{100}}\right]=? \\ $$$${with}\:\left[{x}\right]={greatest}\:{integer}\:{function} \\ $$$$ \\ $$$${can}\:{we}\:{find}\:{a}\:{general}\:{formula}\:{for}\: \\ $$$$\left[\sqrt{\mathrm{1}}\right]+\left[\sqrt{\mathrm{2}}\right]+\left[\sqrt{\mathrm{3}}\right]+...+\left[\sqrt{{n}}\right] \\ $$$${in}\:{terms}\:{of}\:{n}? \\ $$

Question Number 80798    Answers: 0   Comments: 1

Question Number 80795    Answers: 0   Comments: 3

Question Number 80792    Answers: 1   Comments: 0

Π_(n=1) ^∞ [((2n)/(2n−1)).((2n)/(2n+1))] =?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left[\frac{\mathrm{2}{n}}{\mathrm{2}{n}−\mathrm{1}}.\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}\right]\:=? \\ $$

Question Number 80788    Answers: 0   Comments: 0

Question Number 80786    Answers: 1   Comments: 5

Question Number 80780    Answers: 1   Comments: 1

2∙m^x + 3∙n^y = 18 min{ m^x ∙ n^y } = ?

$$\mathrm{2}\centerdot{m}^{{x}} \:+\:\mathrm{3}\centerdot{n}^{{y}} \:\:=\:\:\mathrm{18} \\ $$$${min}\left\{\:{m}^{{x}} \:\centerdot\:{n}^{{y}} \:\right\}\:=\:? \\ $$

Question Number 80777    Answers: 1   Comments: 0

If ^(n+2) C_8 :^(n−2) P_4 = 57 : 16, then the value of n is ......

$$\mathrm{If}\:\:^{{n}+\mathrm{2}} {C}_{\mathrm{8}} \::\:^{{n}−\mathrm{2}} {P}_{\mathrm{4}} =\:\mathrm{57}\::\:\mathrm{16},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{n}\:\mathrm{is}\:...... \\ $$

Question Number 80775    Answers: 0   Comments: 0

There are n straight lines in a plane, no two of which are parallel, and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

$$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane}, \\ $$$$\mathrm{no}\:\mathrm{two}\:\mathrm{of}\:\mathrm{which}\:\mathrm{are}\:\mathrm{parallel},\:\mathrm{and}\:\mathrm{no} \\ $$$$\mathrm{three}\:\mathrm{pass}\:\mathrm{through}\:\mathrm{the}\:\mathrm{same}\:\mathrm{point}. \\ $$$$\mathrm{Their}\:\mathrm{points}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{are}\:\mathrm{joined}. \\ $$$$\mathrm{Then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fresh}\:\mathrm{lines}\:\mathrm{thus}\: \\ $$$$\mathrm{obtained}\:\mathrm{is} \\ $$

Question Number 80770    Answers: 1   Comments: 1

∫x^2 +3x dx=..

$$\int\mathrm{x}^{\mathrm{2}} +\mathrm{3x}\:\mathrm{dx}=.. \\ $$

Question Number 80764    Answers: 1   Comments: 0

show that ∫_0 ^∞ x arctanh(e^(−αx) )dx=((7ζ(3))/(8α^2 ))

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}\:{arctanh}\left({e}^{−\alpha{x}} \right){dx}=\frac{\mathrm{7}\zeta\left(\mathrm{3}\right)}{\mathrm{8}\alpha^{\mathrm{2}} } \\ $$

Question Number 80761    Answers: 0   Comments: 1

Question Number 80760    Answers: 0   Comments: 1

Question Number 80832    Answers: 1   Comments: 2

Identifier les chiffres de l′addition decimale que voici : UN+DOUX+DOUX+DOUX +DOUX=NEUF

$$\boldsymbol{{Identifier}}\:\boldsymbol{{les}}\:\boldsymbol{{chiffres}}\:\boldsymbol{{de}} \\ $$$$\boldsymbol{{l}}'\boldsymbol{{addition}}\:\boldsymbol{{decimale}}\:\boldsymbol{{que}} \\ $$$$\boldsymbol{{voici}}\:: \\ $$$$\boldsymbol{\mathrm{UN}}+\boldsymbol{\mathrm{DOUX}}+\boldsymbol{\mathrm{DOUX}}+\boldsymbol{\mathrm{DOUX}} \\ $$$$+\boldsymbol{\mathrm{DOUX}}=\boldsymbol{\mathrm{NEUF}} \\ $$

Question Number 80752    Answers: 1   Comments: 1

Question Number 80748    Answers: 1   Comments: 3

lim_(x→∞) (((x!)/x^x ))^(1/x) = ?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{x}!}{{x}^{{x}} }\right)^{\frac{\mathrm{1}}{{x}}} \:=\:? \\ $$

Question Number 80747    Answers: 1   Comments: 1

Question Number 80746    Answers: 1   Comments: 2

what is constan term in expansion (1+3x)^5 ((3/x)+1)^2

$${what}\:{is}\:{constan}\:{term}\:{in}\:{expansion} \\ $$$$\left(\mathrm{1}+\mathrm{3}{x}\right)^{\mathrm{5}} \left(\frac{\mathrm{3}}{{x}}+\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 80739    Answers: 0   Comments: 2

cos^3 θ+2sin^2 θ=3 ^ θ∈(0,2π) what is θ ?

$$\mathrm{cos}\:^{\mathrm{3}} \theta+\mathrm{2sin}\:^{\mathrm{2}} \theta=\mathrm{3}\bar {\:}\theta\in\left(\mathrm{0},\mathrm{2}\pi\right) \\ $$$${what}\:{is}\:\theta\:? \\ $$

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