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Question Number 61537 Answers: 2 Comments: 2

Question Number 61536 Answers: 0 Comments: 1

1)let U_n =Σ_(k=0) ^n (−1)^k =1−1+1−1+...(n+1 terms) is lim_(n→+∞) U_n exist ? find U_n by using integr part[..] 2) let V_n = Σ_(k=1) ^n k(−1)^k = −1+2 −3+4+.....(nterms) is lim_(n→+∞) V_n exist find V_n by using integr part[..]

$$\left.\mathrm{1}\right){let}\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:=\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+...\left({n}+\mathrm{1}\:{terms}\right) \\ $$$${is}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} {exist}\:?\:\:{find}\:{U}_{{n}} \:{by}\:{using}\:{integr}\:{part}\left[..\right] \\ $$$$\left.\mathrm{2}\right)\:{let}\:{V}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} \:\:=\:−\mathrm{1}+\mathrm{2}\:−\mathrm{3}+\mathrm{4}+.....\left({nterms}\right) \\ $$$${is}\:{lim}_{{n}\rightarrow+\infty} {V}_{{n}} \:{exist}\: \\ $$$${find}\:{V}_{{n}} {by}\:{using}\:{integr}\:{part}\left[..\right] \\ $$

Question Number 61535 Answers: 0 Comments: 0

calculate ∫_0 ^(π/2) ((ln(1+cosx))/(cosx)) dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{ln}\left(\mathrm{1}+{cosx}\right)}{{cosx}}\:{dx} \\ $$

Question Number 61534 Answers: 0 Comments: 0

calculate f(a) =∫∫_W (x+ay)e^(−x) e^(−ay) dxdy with W_a ={(x,y)∈R^2 /x≥0 ,y≥0 , x+ay ≤1 } a>0

$${calculate}\:{f}\left({a}\right)\:=\int\int_{{W}} \:\left({x}+{ay}\right){e}^{−{x}} \:{e}^{−{ay}} {dxdy}\:{with} \\ $$$${W}_{{a}} =\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\:\:\:,\:{x}+{ay}\:\leqslant\mathrm{1}\:\right\}\:\:\:{a}>\mathrm{0} \\ $$

Question Number 61533 Answers: 0 Comments: 2

∫∫_([0,1]^2 ) ((x−y)/((x^2 +3y^(2 ) +1)^2 )) dxdy

$$\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\:\:\frac{{x}−{y}}{\left({x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}\:} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{dxdy}\: \\ $$

Question Number 61532 Answers: 0 Comments: 0

prove that (((a+b)^n )/(a^n +b^n )) <2^(n−1) ∀ n>1 (n natural)

$${prove}\:{that}\:\:\:\frac{\left({a}+{b}\right)^{{n}} }{{a}^{{n}} \:+{b}^{{n}} }\:<\mathrm{2}^{{n}−\mathrm{1}} \:\:\:\:\:\forall\:{n}>\mathrm{1}\:\:\:\:\left({n}\:{natural}\right) \\ $$

Question Number 61530 Answers: 0 Comments: 5

let U_n =∫_0 ^∞ (x^(−2n) /(1+x^4 )) dx with n integr natural and n≥1 1) calculate U_n interms of n 2) find lim_(n→+∞) n^2 U_n 3) study the serie Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{−\mathrm{2}{n}} }{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:\:\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 61529 Answers: 0 Comments: 0

find ∫_0 ^∞ x^2 e^(−zx^2 ) dx with z from C

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{\mathrm{2}} {e}^{−{zx}^{\mathrm{2}} } {dx}\:\:{with}\:{z}\:{from}\:{C}\: \\ $$

Question Number 61528 Answers: 0 Comments: 4

find ∫_0 ^∞ cos(zx^2 )dx with z ∈ C .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{cos}\left({zx}^{\mathrm{2}} \right){dx}\:{with}\:{z}\:\in\:{C}\:. \\ $$

Question Number 61526 Answers: 0 Comments: 0

Solve for n: D/A×{1−((P×((((1+i)^n ×i)/((1+i)^n −1))))/((P×((((1+i)^r ×i)/((1+i)^r −1))))−(R/i)×[((1/n)+i)×((((1+i)^r ×i)/((1+i)^r −1)))−((1/n)+i)×((((1+i)^n ×i)/((1+i)^n −1)))]))}−1=0

Question Number 61522 Answers: 1 Comments: 0

I=∫((sin x.e^(cos x) −(sin x+cos x)e^((sin x+cos x)) )/(e^(2sin x) −2e^(sin x) +1))dx

$${I}=\int\frac{\mathrm{sin}\:{x}.{e}^{\mathrm{cos}\:{x}} −\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right){e}^{\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)} }{{e}^{\mathrm{2sin}\:{x}} −\mathrm{2}{e}^{\mathrm{sin}\:{x}} +\mathrm{1}}{dx} \\ $$

Question Number 61521 Answers: 1 Comments: 0

Question Number 61520 Answers: 0 Comments: 0

Question Number 61495 Answers: 1 Comments: 7

Question Number 61511 Answers: 1 Comments: 0

cos^(−1) ((2x^2 −1)/(2x^2 )) + cos^(−1) ((x^2 −2)/x^2 )=120°=((2π)/3) Find x

$${cos}^{−\mathrm{1}} \:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\:+\:{cos}^{−\mathrm{1}} \:\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{2}} }=\mathrm{120}°=\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$$\mathrm{Find}\:\:{x} \\ $$

Question Number 61510 Answers: 0 Comments: 0

S_1 =Σ_(k=1) ^n (√((16n−16k)(16n+16k))) S_2 =Σ_(k=1) ^n (√((16k−16)(16k+16))) lim_(n→∞) ((S_1 +S_2 )/n^2 )=?

$${S}_{\mathrm{1}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\sqrt{\left(\mathrm{16}{n}−\mathrm{16}{k}\right)\left(\mathrm{16}{n}+\mathrm{16}{k}\right)} \\ $$$${S}_{\mathrm{2}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\sqrt{\left(\mathrm{16}{k}−\mathrm{16}\right)\left(\mathrm{16}{k}+\mathrm{16}\right)} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{S}_{\mathrm{1}} +{S}_{\mathrm{2}} }{{n}^{\mathrm{2}} }=? \\ $$

Question Number 61490 Answers: 0 Comments: 10

(√(a−(√(a+x))))+(√(a+(√(a−x))))=2x this is the solution Sir Aifour and me found trivial solution a=x=0 a, x ∈R x=((√2)/8)(r+(√(r^2 +4)))(√(2(4a−1)−r^2 −r(√(r^2 +4)))) with r=−2(√((4a−3)/3))sin ((1/3)arcsin ((3(√3))/((4a−3)^(3/2) ))) no solution for a<a_0 with a_0 ≈1.509830340886

$$\sqrt{{a}−\sqrt{{a}+{x}}}+\sqrt{{a}+\sqrt{{a}−{x}}}=\mathrm{2}{x} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{Sir}\:\mathrm{Aifour}\:\mathrm{and}\:\mathrm{me}\:\mathrm{found} \\ $$$$ \\ $$$$\mathrm{trivial}\:\mathrm{solution}\:{a}={x}=\mathrm{0} \\ $$$$ \\ $$$${a},\:{x}\:\in\mathbb{R} \\ $$$$ \\ $$$${x}=\frac{\sqrt{\mathrm{2}}}{\mathrm{8}}\left({r}+\sqrt{{r}^{\mathrm{2}} +\mathrm{4}}\right)\sqrt{\mathrm{2}\left(\mathrm{4}{a}−\mathrm{1}\right)−{r}^{\mathrm{2}} −{r}\sqrt{{r}^{\mathrm{2}} +\mathrm{4}}} \\ $$$$\mathrm{with} \\ $$$${r}=−\mathrm{2}\sqrt{\frac{\mathrm{4}{a}−\mathrm{3}}{\mathrm{3}}}\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{3}}\mathrm{arcsin}\:\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\left(\mathrm{4}{a}−\mathrm{3}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\right) \\ $$$$\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{a}<{a}_{\mathrm{0}} \:\mathrm{with}\:{a}_{\mathrm{0}} \approx\mathrm{1}.\mathrm{509830340886} \\ $$

Question Number 61565 Answers: 2 Comments: 3

x(y+1) + y(x−1) = 12 x(√y) + y(√x) = 21 x, y ∈ R x + y = ?

$${x}\left({y}+\mathrm{1}\right)\:+\:{y}\left({x}−\mathrm{1}\right)\:\:=\:\:\mathrm{12} \\ $$$${x}\sqrt{{y}}\:\:+\:\:{y}\sqrt{{x}}\:\:\:\:=\:\:\mathrm{21} \\ $$$${x},\:{y}\:\:\in\:\:\mathbb{R} \\ $$$${x}\:+\:{y}\:\:=\:\:? \\ $$

Question Number 61479 Answers: 3 Comments: 1

Solve for x: ((6(√(2x)))/(x − 1)) + ((5(√(x − 1)))/(2x)) = 13

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}:\:\:\:\:\:\:\:\frac{\mathrm{6}\sqrt{\mathrm{2x}}}{\mathrm{x}\:−\:\mathrm{1}}\:+\:\frac{\mathrm{5}\sqrt{\mathrm{x}\:−\:\mathrm{1}}}{\mathrm{2x}}\:\:\:=\:\:\mathrm{13} \\ $$

Question Number 61470 Answers: 1 Comments: 0

Is there any other solution besides {x=a,y=b} or {x=b,y=a} of the following system of equations x+y=a+b ∧ x^7 +y^7 =a^7 +b^7 ?

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{other}\:\mathrm{solution}\:\mathrm{besides} \\ $$$$\left\{\mathrm{x}=\mathrm{a},\mathrm{y}=\mathrm{b}\right\}\:\mathrm{or}\:\left\{\mathrm{x}=\mathrm{b},\mathrm{y}=\mathrm{a}\right\}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\:\:\:\:\mathrm{x}+\mathrm{y}=\mathrm{a}+\mathrm{b}\:\:\wedge\:\mathrm{x}^{\mathrm{7}} +\mathrm{y}^{\mathrm{7}} =\mathrm{a}^{\mathrm{7}} +\mathrm{b}^{\mathrm{7}} \:\:? \\ $$$$ \\ $$

Question Number 61465 Answers: 1 Comments: 1

∫_0 ^(2π) (1/(a^2 cos^2 (t)+b^2 sin^2 (t)))dt=((2π)/(ab))?