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Question Number 94660    Answers: 0   Comments: 0

u_n =(1+(1/n^2 ))(1+(2/n^2 ))...(1+(n/n^2 )) find lim_(n→+∞) u_n

$${u}_{{n}} =\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\left(\mathrm{1}+\frac{\mathrm{2}}{{n}^{\mathrm{2}} }\right)...\left(\mathrm{1}+\frac{{n}}{{n}^{\mathrm{2}} }\right) \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \\ $$

Question Number 94659    Answers: 1   Comments: 0

calculate lim_(n→+∞) ∫_0 ^∞ (1−(t/n))^n e^(−3t) dt

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{\infty} \:\left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} \:{e}^{−\mathrm{3}{t}} \:{dt} \\ $$

Question Number 94658    Answers: 0   Comments: 0

find lim_(n→+∞) ∫_0 ^1 (1−(t/n))^n arctan(1+nt)dt

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} \:{arctan}\left(\mathrm{1}+{nt}\right){dt} \\ $$

Question Number 94657    Answers: 0   Comments: 0

let u_n =Σ_(k=1) ^n (1/((^p (√k)))) detetmine a equivalent of u_n (n→+∞)

$${let}\:{u}_{{n}} \:=\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{\left(^{{p}} \sqrt{{k}}\right)} \\ $$$${detetmine}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \\ $$$$\left({n}\rightarrow+\infty\right) \\ $$$$ \\ $$

Question Number 94655    Answers: 0   Comments: 0

solve (1+nz)^p +(1−nz)^p =0 n and p ontegr natural

$${solve}\:\left(\mathrm{1}+{nz}\right)^{{p}} +\left(\mathrm{1}−{nz}\right)^{{p}} \:=\mathrm{0} \\ $$$${n}\:{and}\:{p}\:{ontegr}\:{natural} \\ $$

Question Number 94654    Answers: 0   Comments: 0

prove that Σ_(i=1) ^n x_i y_i ≤(Σ_(i=1) ^n x_i ^2 )^(1/2) (Σ_(i=1) ^n y_i ^2 )^(1/2) x_i and y_i reals ≥0

$${prove}\:{that}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}_{{i}} {y}_{{i}} \leqslant\left(\sum_{{i}=\mathrm{1}} ^{{n}} {x}_{{i}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\sum_{{i}=\mathrm{1}} ^{{n}} {y}_{{i}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$${x}_{{i}} \:{and}\:{y}_{{i}} \:{reals}\:\geqslant\mathrm{0} \\ $$

Question Number 94652    Answers: 0   Comments: 0

find lim_(ξ→0) ∫_0 ^1 ((sin(ξx))/((√(1+ξx^2 ))−(√(1−ξx^2 ))))dx

$${find}\:{lim}_{\xi\rightarrow\mathrm{0}} \:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{sin}\left(\xi{x}\right)}{\sqrt{\mathrm{1}+\xi{x}^{\mathrm{2}} }−\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 94650    Answers: 1   Comments: 1

let f(x) =(x+1)^9 e^(−3x) calculstr f^((7)) (0) and f^((5)) (1)

$${let}\:{f}\left({x}\right)\:=\left({x}+\mathrm{1}\right)^{\mathrm{9}} \:{e}^{−\mathrm{3}{x}} \\ $$$${calculstr}\:{f}^{\left(\mathrm{7}\right)} \left(\mathrm{0}\right)\:{and}\:{f}^{\left(\mathrm{5}\right)} \left(\mathrm{1}\right) \\ $$

Question Number 94649    Answers: 0   Comments: 0

calculate ∫_0 ^∞ e^(−x) lnx dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {lnx}\:{dx} \\ $$

Question Number 94648    Answers: 0   Comments: 0

find a equivslent for u_n =1 +2^α +3^α +....+n^α n→+∞ (α>0)

$${find}\:{a}\:{equivslent}\:{for} \\ $$$${u}_{{n}} =\mathrm{1}\:+\mathrm{2}^{\alpha} \:+\mathrm{3}^{\alpha} \:+....+{n}^{\alpha} \\ $$$${n}\rightarrow+\infty\:\:\:\:\:\:\:\:\:\left(\alpha>\mathrm{0}\right) \\ $$

Question Number 94635    Answers: 1   Comments: 0

A father with 8 children takes them 3 at a time to the Gardens, as often as he can without taking the same 3 children together more than once. The number of times each child will go to the garden is

$$\mathrm{A}\:\mathrm{father}\:\mathrm{with}\:\mathrm{8}\:\mathrm{children}\:\mathrm{takes}\:\mathrm{them} \\ $$$$\mathrm{3}\:\mathrm{at}\:\mathrm{a}\:\mathrm{time}\:\mathrm{to}\:\mathrm{the}\:\mathrm{Gardens},\:\mathrm{as}\:\mathrm{often}\:\mathrm{as} \\ $$$$\mathrm{he}\:\mathrm{can}\:\mathrm{without}\:\mathrm{taking}\:\mathrm{the}\:\mathrm{same}\:\mathrm{3}\:\mathrm{children} \\ $$$$\mathrm{together}\:\mathrm{more}\:\mathrm{than}\:\mathrm{once}.\:\mathrm{The}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{times}\:\mathrm{each}\:\mathrm{child}\:\mathrm{will}\:\mathrm{go}\:\mathrm{to}\:\mathrm{the}\:\mathrm{garden} \\ $$$$\mathrm{is} \\ $$

Question Number 94632    Answers: 0   Comments: 1

The coefficient of x^m and x^n (m, n ∈ N) in the expansion of (1+x)^(m+n) are

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{m}} \:\mathrm{and}\:{x}^{{n}} \left({m},\:{n}\:\in\:{N}\right)\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{{m}+{n}} \:\mathrm{are} \\ $$

Question Number 94629    Answers: 1   Comments: 0

In △ABC, a=(√3)+1, B=30°, C=45°, then c = ____.

$$\mathrm{In}\:\bigtriangleup{ABC},\:{a}=\sqrt{\mathrm{3}}+\mathrm{1},\:{B}=\mathrm{30}°,\:{C}=\mathrm{45}°, \\ $$$$\mathrm{then}\:{c}\:=\:\_\_\_\_. \\ $$

Question Number 94628    Answers: 1   Comments: 0

The perimeter of a △ ABC is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

$$\mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{a}\:\bigtriangleup\:{ABC}\:\mathrm{is}\:\mathrm{6}\:\mathrm{times} \\ $$$$\mathrm{the}\:\mathrm{arithmetic}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sines}\:\mathrm{of} \\ $$$$\mathrm{its}\:\mathrm{angles}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{side}\:{a}\:\mathrm{is}\:\:\mathrm{1},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{angle}\:{A}\:\:\mathrm{is} \\ $$

Question Number 94625    Answers: 0   Comments: 3

∫_0 ^1 ln (Γ(x) dx =? note Γ(x) :Gamma function

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{ln}\:\left(\Gamma\left({x}\right)\:{dx}\:=?\right. \\ $$$${note}\:\Gamma\left({x}\right)\::\mathrm{Gamma}\:\mathrm{function} \\ $$

Question Number 94622    Answers: 0   Comments: 0

Question Number 94616    Answers: 1   Comments: 0

S=(√(((fx^(^ 2) )/n)−(((fx)/n))^(^ 2) ))

$${S}=\sqrt{\frac{{f}\overset{\hat {}\mathrm{2}} {{x}}}{{n}}−\left(\frac{{fx}}{{n}}\overset{\hat {}\mathrm{2}} {\right)}} \\ $$

Question Number 94609    Answers: 2   Comments: 0

∫((x^2 −1)/((√(x+1))+(√(2x+3))))dx

$$\int\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\sqrt{{x}+\mathrm{1}}+\sqrt{\mathrm{2}{x}+\mathrm{3}}}{dx} \\ $$

Question Number 94613    Answers: 1   Comments: 3

Question Number 94603    Answers: 1   Comments: 1

List the elements in C={x:x is an x^2 ≤4, integer}

$$\mathrm{List}\:\mathrm{the}\:\mathrm{elements}\:\mathrm{in}\: \\ $$$${C}=\left\{{x}:{x}\:\mathrm{is}\:\mathrm{an}\:{x}^{\mathrm{2}} \leqslant\mathrm{4},\:\mathrm{integer}\right\} \\ $$

Question Number 94602    Answers: 1   Comments: 0

Question Number 94601    Answers: 1   Comments: 0

Question Number 94589    Answers: 0   Comments: 2

covert the point(p,θ,ϕ)=[4(√3),(π/6),(π/3)] to cartesian coordinates.

$${covert}\:{the}\:{point}\left({p},\theta,\varphi\right)=\left[\mathrm{4}\sqrt{\mathrm{3}},\frac{\pi}{\mathrm{6}},\frac{\pi}{\mathrm{3}}\right]\: \\ $$$${to}\:{cartesian}\:{coordinates}. \\ $$

Question Number 94581    Answers: 0   Comments: 2

Question Number 94573    Answers: 1   Comments: 2

Given a, b, and c, 3 real numbers which satisfy the equation { ((a+b+c=312)),((c+a=192)) :} Find these real numbers such that they form 3 consecutive terms of a Geometric Progression.

$$\mathrm{Given}\:\mathrm{a},\:\mathrm{b},\:\mathrm{and}\:\mathrm{c},\:\mathrm{3}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{which}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\begin{cases}{\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{312}}\\{\mathrm{c}+\mathrm{a}=\mathrm{192}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{these}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{they}\:\mathrm{form} \\ $$$$\mathrm{3}\:\mathrm{consecutive}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{a}\:\mathrm{Geometric}\:\mathrm{Progression}. \\ $$

Question Number 94572    Answers: 0   Comments: 0

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