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Question Number 197359 Answers: 2 Comments: 1

lim_(x→0) ((1−cosxcos2x...cos(nx))/x^2 ) = ((n(n+1)(2n+1))/(12))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{1}−{cosxcos}\mathrm{2}{x}...{cos}\left({nx}\right)}{{x}^{\mathrm{2}} }\:=\:\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{12}}\: \\ $$

Question Number 197349 Answers: 1 Comments: 0

calcul Σ_(n=1) ^(+oo) (−1)^(n ) ((2n+1)/(n(n+1)))

$${calcul}\: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{+{oo}} {\sum}}\left(−\mathrm{1}\right)^{{n}\:} \frac{\mathrm{2}{n}+\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)} \\ $$

Question Number 197346 Answers: 1 Comments: 0

Question Number 197345 Answers: 1 Comments: 0

Question Number 197344 Answers: 0 Comments: 1

((d )/dt)∙(dx^𝛌 /dt)+(1/2)g^(𝛌𝛂) (∂_𝛍 ^ g_(𝛂𝛎) +∂_𝛎 ^ g_(𝛂𝛍) −∂_𝛂 ^ g_(𝛍𝛎) )(dx^𝛍 /dt)∙(dx^𝛎 /dt)=0

$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{\boldsymbol{\lambda}} }{\mathrm{d}{t}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{g}^{\boldsymbol{\lambda\alpha}} \left(\partial_{\boldsymbol{\mu}} ^{\:} \mathrm{g}_{\boldsymbol{\alpha\nu}} +\partial_{\boldsymbol{\nu}} ^{\:} \mathrm{g}_{\boldsymbol{\alpha\mu}} −\partial_{\boldsymbol{\alpha}} ^{\:} \mathrm{g}_{\boldsymbol{\mu\nu}} \right)\frac{\mathrm{d}{x}^{\boldsymbol{\mu}} }{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{\boldsymbol{\nu}} }{\mathrm{d}{t}}=\mathrm{0} \\ $$

Question Number 197343 Answers: 0 Comments: 0

calculate ∫_0 ^(π/2) ln(cosx).ln(sinx)dx

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

Question Number 197338 Answers: 1 Comments: 0

∫(x^2 /(x^2 +1))dx

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

Question Number 197336 Answers: 2 Comments: 0

lim_(x→+∞) ((1/x^2 )+cosx)=?

$${lim}_{{x}\rightarrow+\infty} \left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\mathrm{cos}{x}\right)=? \\ $$

Question Number 197327 Answers: 2 Comments: 0

trigonometry... P = Π_(k=1) ^(44) ( 1 + tan(k) ) = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{trigonometry}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{P}\:=\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{44}} {\prod}}\left(\:\:\mathrm{1}\:+\:{tan}\left({k}\right)\:\right)\:=\:?\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 197335 Answers: 1 Comments: 0

Question Number 197325 Answers: 1 Comments: 0

Question Number 197323 Answers: 1 Comments: 0

If f(x)=((sin(x))/x) and S_n (α)=Σ_(k=1) ^n [f(kπ+(π/α))+f(kπ−(π/α))] (α>1) Prove that lim_(n→+∞) S_n (α)=1−f((π/α))

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{x}}\:\:\:\mathrm{and}\:\mathrm{S}_{\mathrm{n}} \left(\alpha\right)=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left[\mathrm{f}\left(\mathrm{k}\pi+\frac{\pi}{\alpha}\right)+\mathrm{f}\left(\mathrm{k}\pi−\frac{\pi}{\alpha}\right)\right]\:\:\:\:\left(\alpha>\mathrm{1}\right) \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow+\infty} {\:\mathrm{lim}}\:\mathrm{S}_{\mathrm{n}} \left(\alpha\right)=\mathrm{1}−\mathrm{f}\left(\frac{\pi}{\alpha}\right) \\ $$

Question Number 197320 Answers: 1 Comments: 0

Σ_(k=1) ^n (−1)^(k(k+1))

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}\left({k}+\mathrm{1}\right)} \\ $$

Question Number 197317 Answers: 0 Comments: 0

if x = ((cos θ)/u) , y = ((sin θ)/u) and z = f(x,y) then show that (∂^2 z/∂x^2 ) + (∂^2 z/∂y^2 ) = u^4 (∂^2 z/∂u^2 ) + u^3 (∂z/∂u) + u^4 (∂^2 z/∂θ^2 )

$$\:\mathrm{if}\:\:\:\mathrm{x}\:\:=\:\:\frac{\mathrm{cos}\:\theta}{\mathrm{u}}\:\:,\:\mathrm{y}\:\:=\:\frac{\mathrm{sin}\:\theta}{\mathrm{u}}\:\:{and}\:\mathrm{z}\:\:=\:\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{x}^{\mathrm{2}} }\:+\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{y}^{\mathrm{2}} }\:=\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{u}^{\mathrm{2}} }\:+\:\mathrm{u}^{\mathrm{3}} \:\frac{\partial\mathrm{z}}{\partial\mathrm{u}}\:+\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\theta^{\mathrm{2}} } \\ $$

Question Number 197312 Answers: 1 Comments: 0

(((log_2 20)^2 −(log_2 5)^2 )/(log_2 10))=?

$$\frac{\left({log}_{\mathrm{2}} \mathrm{20}\right)^{\mathrm{2}} −\left({log}_{\mathrm{2}} \mathrm{5}\right)^{\mathrm{2}} }{{log}_{\mathrm{2}} \mathrm{10}}=? \\ $$

Question Number 197311 Answers: 1 Comments: 0

Prove that _(n+1) C_r = _n C_r + _n C_(r−1)

$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:\:_{\mathrm{n}+\mathrm{1}} \:\mathrm{C}_{\mathrm{r}} \:=\:_{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \:+\:_{\mathrm{n}} \mathrm{C}_{\mathrm{r}−\mathrm{1}} \: \\ $$

Question Number 197310 Answers: 1 Comments: 0

Question Number 197301 Answers: 1 Comments: 0

how do i calculate this lim_(x→-∞) ((x^4 +2x^2 +x−2)/(x^3 +2x^2 +x−1)) multiplying both numerator and denumerator by (1/x^4 ) lim_(x→-∞) ((1+(2/x^2 )+(1/x^3 )−(2/x^4 ))/((1/x)+(2/x^2 )+(1/x^3 )−(1/x^4 ))) ((1+0+0−0)/(0+0+0−0)) ∞ which is not true the answer is -∞, i tried multiplying (1/x^3 ) and got -∞ but still confused what did i do wrong using (1/x^4 )

$$ \\ $$$$\:{how}\:{do}\:{i}\:{calculate}\:{this} \\ $$$$\:\underset{{x}\rightarrow-\infty} {\mathrm{lim}}\:\frac{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} +{x}−\mathrm{2}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +{x}−\mathrm{1}} \\ $$$$\:{multiplying}\:{both}\:{numerator} \\ $$$$\:{and}\:{denumerator}\:{by}\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$$$\:\underset{{x}\rightarrow-\infty} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }−\frac{\mathrm{2}}{{x}^{\mathrm{4}} }}{\frac{\mathrm{1}}{{x}}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }−\frac{\mathrm{1}}{{x}^{\mathrm{4}} }} \\ $$$$\:\frac{\mathrm{1}+\mathrm{0}+\mathrm{0}−\mathrm{0}}{\mathrm{0}+\mathrm{0}+\mathrm{0}−\mathrm{0}} \\ $$$$\:\infty \\ $$$$\:{which}\:{is}\:{not}\:{true}\:{the}\:{answer}\:{is}\:-\infty, \\ $$$$\:{i}\:{tried}\:{multiplying}\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:{and}\:{got}\:-\infty \\ $$$$\:{but}\:{still}\:{confused}\:{what}\:{did}\:{i}\:{do}\:{wrong} \\ $$$$\:{using}\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$$$ \\ $$

Question Number 197299 Answers: 1 Comments: 0

((−64))^(1/6) −((−10))^(1/(10)) =? I need so much plz

$$\sqrt[{\mathrm{6}}]{−\mathrm{64}}−\sqrt[{\mathrm{10}}]{−\mathrm{10}}=? \\ $$$$\boldsymbol{{I}}\:\boldsymbol{\mathrm{need}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{much}}\:\boldsymbol{\mathrm{plz}} \\ $$

Question Number 197292 Answers: 2 Comments: 0

lim_(n→∞) ∫_(0 ) ^1 ((nx^(n−1) )/(1+x))dx = ?

$$\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{{nx}^{{n}−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:\:=\:\:\:? \\ $$

Question Number 197290 Answers: 0 Comments: 1