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AllQuestion and Answers: Page 778

Question Number 141380 Answers: 0 Comments: 0

Question Number 141378 Answers: 2 Comments: 0

prove that:: Π_(n=0) ^∞ (((5n+2)(5n+3))/((5n+1)(5n+4))) =ϕ ϕ:= ((1+(√5))/2)

$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\frac{\left(\mathrm{5}{n}+\mathrm{2}\right)\left(\mathrm{5}{n}+\mathrm{3}\right)}{\left(\mathrm{5}{n}+\mathrm{1}\right)\left(\mathrm{5}{n}+\mathrm{4}\right)}\:=\varphi\: \\ $$$$\:\:\:\:\:\:\:\varphi:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$

Question Number 141294 Answers: 5 Comments: 0

Find max & min value of f(x)=(x/(x^2 −5x+9)).

$${Find}\:{max}\:\&\:{min}\:{value}\:{of} \\ $$$$\:{f}\left({x}\right)=\frac{{x}}{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{9}}. \\ $$

Question Number 141328 Answers: 2 Comments: 0

...... Evaluate: F :=Σ_(n=2) ^∞ (((−1)^n ζ(n))/(n+1)) =? .......

$$......\:{Evaluate}: \\ $$$$\:\:\:\:\:\mathscr{F}\::=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{{n}+\mathrm{1}}\:=? \\ $$$$....... \\ $$

Question Number 141312 Answers: 0 Comments: 0

Σ_(n=1) ^∞ (−1)^n ((Si(2πn)−(π/2))/n)=?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{{Si}\left(\mathrm{2}\pi{n}\right)−\frac{\pi}{\mathrm{2}}}{{n}}=? \\ $$

Question Number 141289 Answers: 1 Comments: 1

Question Number 141381 Answers: 0 Comments: 0

Let a,b ≥ 0 . Prove that (a+b+2)^3 ≥ ((27)/2)(a^2 +ab+b^2 )

$$\mathrm{Let}\:\:{a},{b}\:\geqslant\:\mathrm{0}\:.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+{b}+\mathrm{2}\right)^{\mathrm{3}} \:\geqslant\:\frac{\mathrm{27}}{\mathrm{2}}\left({a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 141291 Answers: 0 Comments: 3

Question Number 141269 Answers: 1 Comments: 3

θ+φ+ψ=π (angles of a △) find maximum of (φ−θ)^2 +(ψ−φ)^2 +(ψ−θ)^2 .

$$\theta+\phi+\psi=\pi\:\:\left({angles}\:{of}\:{a}\:\bigtriangleup\right) \\ $$$${find}\:{maximum}\:{of} \\ $$$$\:\left(\phi−\theta\right)^{\mathrm{2}} +\left(\psi−\phi\right)^{\mathrm{2}} +\left(\psi−\theta\right)^{\mathrm{2}} \:. \\ $$

Question Number 141412 Answers: 1 Comments: 2

Find the range of real number of q such that the function f(x) = cos x(q sin^2 x−5) have minimum value is −5 .

$$\:{Find}\:{the}\:{range}\:{of}\:{real}\:{number} \\ $$$${of}\:{q}\:{such}\:{that}\:{the}\:{function}\: \\ $$$$\:{f}\left({x}\right)\:=\:\mathrm{cos}\:{x}\left({q}\:\mathrm{sin}\:^{\mathrm{2}} {x}−\mathrm{5}\right)\:{have} \\ $$$${minimum}\:{value}\:{is}\:−\mathrm{5}\:. \\ $$

Question Number 141283 Answers: 0 Comments: 0

((ζ(2))/2^3 )−((ζ(3))/3^3 )+((ζ(4))/4^3 )−((ζ(5))/5^3 )+...

$$\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}^{\mathrm{3}} }−\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{3}^{\mathrm{3}} }+\frac{\zeta\left(\mathrm{4}\right)}{\mathrm{4}^{\mathrm{3}} }−\frac{\zeta\left(\mathrm{5}\right)}{\mathrm{5}^{\mathrm{3}} }+... \\ $$

Question Number 141257 Answers: 0 Comments: 0

Question Number 141252 Answers: 1 Comments: 0

Question Number 141249 Answers: 2 Comments: 1

Question Number 141246 Answers: 1 Comments: 0

r=q+1 pq=q+1 c^2 p=qr^2 , help find p, q, r.

$${r}={q}+\mathrm{1} \\ $$$${pq}={q}+\mathrm{1} \\ $$$${c}^{\mathrm{2}} {p}={qr}^{\mathrm{2}} \:\:\:,\:{help}\:{find}\:{p},\:{q},\:{r}. \\ $$

Question Number 141244 Answers: 1 Comments: 1

I=∫_0 ^( ∞) ((x{(a^2 −b^2 )x−2a^2 x^2 −2b^2 }dx)/((a^2 x^2 +b^2 )^2 {(a^2 −b^2 )x+a^2 x^2 +b^2 }))

$${I}=\int_{\mathrm{0}} ^{\:\infty} \frac{{x}\left\{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right){x}−\mathrm{2}{a}^{\mathrm{2}} {x}^{\mathrm{2}} −\mathrm{2}{b}^{\mathrm{2}} \right\}{dx}}{\left({a}^{\mathrm{2}} {x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{2}} \left\{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right){x}+{a}^{\mathrm{2}} {x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right\}} \\ $$

Question Number 141240 Answers: 0 Comments: 0

Solve using fourier′s series −y′′+y=e^(−2∣x∣)

$$\mathrm{Solve}\:\mathrm{using}\:\mathrm{fourier}'\mathrm{s}\:\mathrm{series} \\ $$$$−\mathrm{y}''+\mathrm{y}=\mathrm{e}^{−\mathrm{2}\mid\mathrm{x}\mid} \\ $$

Question Number 141227 Answers: 1 Comments: 2

Question Number 141243 Answers: 0 Comments: 0

{ ((ty′′+y′+ty=0)),((y(0^+ )=1)) :}

$$\begin{cases}{\mathrm{ty}''+\mathrm{y}'+\mathrm{ty}=\mathrm{0}}\\{\mathrm{y}\left(\mathrm{0}^{+} \right)=\mathrm{1}}\end{cases} \\ $$

Question Number 141222 Answers: 1 Comments: 0

calculate ∫_0 ^(4π) (dθ/((x^2 +2x cosθ +1)^2 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{4}\pi} \:\:\frac{\mathrm{d}\theta}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}\:\mathrm{cos}\theta\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 141221 Answers: 2 Comments: 2

Question Number 141220 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((logx)/(1+x^4 )) dx

$$\mathrm{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{logx}}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\:\mathrm{dx} \\ $$

Question Number 141219 Answers: 1 Comments: 0

find ∫∫_([0,1]^2 ) e^(−(x^2 +y^2 )) (√(x^4 +y^4 ))dxdy

$$\mathrm{find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\mathrm{e}^{−\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)} \sqrt{\mathrm{x}^{\mathrm{4}} +\mathrm{y}^{\mathrm{4}} }\mathrm{dxdy} \\ $$

Question Number 141218 Answers: 3 Comments: 0

calculate I =∫_0 ^(π/2) ^4 (√(tanx))log(tanx)dx and J =∫_0 ^(π/2) ((log(tanx))/((^3 (√(tanx)))))dx

$$\mathrm{calculate}\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:^{\mathrm{4}} \sqrt{\mathrm{tanx}}\mathrm{log}\left(\mathrm{tanx}\right)\mathrm{dx}\:\:\mathrm{and} \\ $$$$\mathrm{J}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{log}\left(\mathrm{tanx}\right)}{\left(^{\mathrm{3}} \sqrt{\mathrm{tanx}}\right)}\mathrm{dx} \\ $$

Question Number 141198 Answers: 0 Comments: 2

Question Number 141197 Answers: 1 Comments: 0

Write the next three terms 1, (3/7), (8/3), ___, ___, ___, ...

$${Write}\:{the}\:{next}\:{three}\:{terms}\:\mathrm{1},\:\frac{\mathrm{3}}{\mathrm{7}},\:\frac{\mathrm{8}}{\mathrm{3}},\:\_\_\_,\:\_\_\_,\:\_\_\_,\:... \\ $$

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