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AlgebraQuestion and Answers: Page 65

Question Number 196024 Answers: 1 Comments: 0

lim_(x→0^+ ) [xΣ_(n=1) ^∞ ((1/n^(x+1) ))]=λ , evalute λ

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[{x}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}^{{x}+\mathrm{1}} }\right)\right]=\lambda\:,\:{evalute}\:\lambda \\ $$

Question Number 196015 Answers: 0 Comments: 1

Question Number 195998 Answers: 0 Comments: 0

We can transform to get rid of the ((...))^(1/3) a^(1/3) +b^(1/3) =c^(1/3) (a^(1/3) +b^(1/3) )^3 =c a+b+3a^(1/3) b^(1/3) (a^(1/3) +b^(1/3) )=c 3a^(1/3) b^(1/3) c^(1/3) =c−a−b 27abc=(c−a−b)^3 Is it possible to do the same for a^(1/3) +b^(1/3) =c^(1/3) +d^(1/3) I found no path yet...

Question Number 195971 Answers: 3 Comments: 0

if f′(x)=((f(x+a)−f(x))/a), find f(x).

$${if}\:{f}'\left({x}\right)=\frac{{f}\left({x}+{a}\right)−{f}\left({x}\right)}{{a}},\:{find}\:{f}\left({x}\right). \\ $$

Question Number 195911 Answers: 0 Comments: 0

Question Number 195904 Answers: 1 Comments: 0

Σ_(n=0) ^∞ [((2^n (2n)!)/(3^(2n+1) (n+1)!n!))]=λ Evaluate (λ)

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\mathrm{2}^{{n}} \left(\mathrm{2}{n}\right)!}{\mathrm{3}^{\mathrm{2}{n}+\mathrm{1}} \left({n}+\mathrm{1}\right)!{n}!}\right]=\lambda \\ $$$${Evaluate}\:\left(\lambda\right) \\ $$

Question Number 195931 Answers: 1 Comments: 2

Question Number 195898 Answers: 1 Comments: 0

Question Number 195855 Answers: 1 Comments: 0

{ ((3(√(((12))^(1/3) −(3)^(1/3) ))=(x)^(1/3) +(y)^(1/3) −(z)^(1/3) )),((x,y,z ∈ N)) :} ⇒ x,y,z =? please help me

$$\begin{cases}{\mathrm{3}\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{12}}−\sqrt[{\mathrm{3}}]{\mathrm{3}}}=\sqrt[{\mathrm{3}}]{{x}}+\sqrt[{\mathrm{3}}]{{y}}−\sqrt[{\mathrm{3}}]{{z}}}\\{{x},{y},{z}\:\in\:{N}}\end{cases}\:\:\Rightarrow\:{x},{y},{z}\:=? \\ $$$${please}\:{help}\:{me} \\ $$

Question Number 195820 Answers: 1 Comments: 0

a,b,c>0 &abc=1,prove that (1/(1+a+b))+(1/(1+b+c))+(1/(1+c+a))≤1

$${a},{b},{c}>\mathrm{0}\:\&{abc}=\mathrm{1},{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+{a}+{b}}+\frac{\mathrm{1}}{\mathrm{1}+{b}+{c}}+\frac{\mathrm{1}}{\mathrm{1}+{c}+{a}}\leqslant\mathrm{1} \\ $$

Question Number 195813 Answers: 0 Comments: 0

{ ((3(√(((12))^(1/3) −(3)^(1/3) )) = (x)^(1/3) + (y)^(1/3) −(z)^(1/3) )),((x,y,z ∈ N)) :} ⇒ x,y,z =? mr.W please help me and other my friends please help me

$$\begin{cases}{\mathrm{3}\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{12}}−\sqrt[{\mathrm{3}}]{\mathrm{3}}}\:\:=\:\sqrt[{\mathrm{3}}]{{x}}\:+\:\sqrt[{\mathrm{3}}]{{y}}\:−\sqrt[{\mathrm{3}}]{{z}}}\\{{x},{y},{z}\:\in\:{N}}\end{cases}\:\Rightarrow\:{x},{y},{z}\:=? \\ $$$${mr}.{W}\:{please}\:{help}\:{me} \\ $$$${and}\:{other}\:{my}\:{friends}\:{please}\:{help}\:{me} \\ $$

Question Number 195809 Answers: 2 Comments: 3

if x^5 +x+1=0, find x^3 −x^2 =?

$${if}\:{x}^{\mathrm{5}} +{x}+\mathrm{1}=\mathrm{0},\:{find}\:{x}^{\mathrm{3}} −{x}^{\mathrm{2}} =? \\ $$

Question Number 195765 Answers: 2 Comments: 0

hello { ((x^3 +(1/x^3 ) = 18)),((x>1)) :} ⇒ x^5 −(1/x^5 ) = ?

$${hello} \\ $$$$ \\ $$$$\:\begin{cases}{{x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:=\:\mathrm{18}}\\{{x}>\mathrm{1}}\end{cases}\:\:\Rightarrow\:\:\:{x}^{\mathrm{5}} −\frac{\mathrm{1}}{{x}^{\mathrm{5}} }\:=\:? \\ $$

Question Number 195733 Answers: 1 Comments: 0

prove that Σ_(n=2) ^∞ [(B_n^_ /((n−2)!))]=((e(3−e))/((e−1)^3 )) where B_n^_ is the n− th bernouli′s number

$${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\left[\frac{{B}_{\overset{\_} {{n}}} }{\left({n}−\mathrm{2}\right)!}\right]=\frac{{e}\left(\mathrm{3}−{e}\right)}{\left({e}−\mathrm{1}\right)^{\mathrm{3}} } \\ $$$${where}\:{B}_{\overset{\_} {{n}}} \:{is}\:{the}\:{n}−\:{th}\:{bernouli}'{s}\:{number} \\ $$

Question Number 195693 Answers: 2 Comments: 1

hello [Σ_(n=1) ^(10000) (1/( (√n)))]=? [ ] : is bracket thank you

$${hello} \\ $$$$\left[\underset{{n}=\mathrm{1}} {\overset{\mathrm{10000}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{n}}}\right]=? \\ $$$$\left[\:\right]\::\:{is}\:{bracket} \\ $$$${thank}\:{you} \\ $$$$ \\ $$

Question Number 195680 Answers: 0 Comments: 0

Question Number 195653 Answers: 2 Comments: 0

_( →0) ( 1)^((√3)/ ) =?

$$\underset{ \rightarrow\mathrm{0}} { }\left( \mathrm{1}\right)^{\frac{\sqrt{\mathrm{3}}}{ }} \:=? \\ $$$$ \\ $$

Question Number 195652 Answers: 2 Comments: 0

e^(x+y) −e^(x−y) =1 then find (dy/dx)=?

$$ \\ $$$$\mathrm{e}^{\mathrm{x}+\mathrm{y}} −\mathrm{e}^{\mathrm{x}−\mathrm{y}} =\mathrm{1} \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}=? \\ $$$$ \\ $$$$ \\ $$

Question Number 195651 Answers: 1 Comments: 0

0<x<1 (1/(1+x^1 ))+((2x)/(1+x^2 ))+((4x^3 )/(1+x^4 ))+((8x^7 )/(1+x^8 ))+((16x^(15) )/(1+x^(16) ))+....+∞ evaluate the previous summation

$$\mathrm{0}<{x}<\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{1}} }+\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }+\frac{\mathrm{4}{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{4}} }+\frac{\mathrm{8}{x}^{\mathrm{7}} }{\mathrm{1}+{x}^{\mathrm{8}} }+\frac{\mathrm{16}{x}^{\mathrm{15}} }{\mathrm{1}+{x}^{\mathrm{16}} }+....+\infty \\ $$$${evaluate}\:{the}\:{previous}\:{summation} \\ $$

Question Number 195647 Answers: 1 Comments: 0

f(x)=((1276)/((x−1)^(ln(2/(4589))) )) domain f(x)=?

$${f}\left({x}\right)=\frac{\mathrm{1276}}{\left({x}−\mathrm{1}\right)^{{ln}\frac{\mathrm{2}}{\mathrm{4589}}} } \\ $$$${domain}\:{f}\left({x}\right)=? \\ $$

Question Number 195689 Answers: 0 Comments: 0

∫_1 ^3 f(x)^3 f′(x)dx=∫_1 ^3 (f(x)^3 )df(x)=∫_1 ^3 t^3 dt=(t^4 /4)∫_1 ^3 =((f(x)^4 )/4)∫_1 3

$$\int_{\mathrm{1}} ^{\mathrm{3}} {f}\left({x}\right)^{\mathrm{3}} {f}'\left({x}\right){dx}=\int_{\mathrm{1}} ^{\mathrm{3}} \left({f}\left({x}\right)^{\mathrm{3}} \right){df}\left({x}\right)=\int_{\mathrm{1}} ^{\mathrm{3}} {t}^{\mathrm{3}} {dt}=\frac{{t}^{\mathrm{4}} }{\mathrm{4}}\int_{\mathrm{1}} ^{\mathrm{3}} =\frac{{f}\left({x}\right)^{\mathrm{4}} }{\mathrm{4}}\int_{\mathrm{1}} \mathrm{3} \\ $$

Question Number 195628 Answers: 2 Comments: 3

an unsolved old question #190875 a, b, c are real roots of the equation x^3 −7x^2 +4x+1=0. find (1/( (a)^(1/3) ))+(1/( (b)^(1/3) ))+(1/( (c)^(1/3) ))=?

$$\underline{{an}\:{unsolved}\:{old}\:{question}\:#\mathrm{190875}} \\ $$$${a},\:{b},\:{c}\:{are}\:{real}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${x}^{\mathrm{3}} −\mathrm{7}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{1}=\mathrm{0}. \\ $$$${find}\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{a}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{b}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{c}}}=? \\ $$

Question Number 195571 Answers: 2 Comments: 0

let f(x+y)+f(x−y)=2f(x)f(y)∧f((1/2))=−1 compute Σ_(k=1) ^(20) [(1/(sin (k)sin (k+f(k))))]

$${let}\:{f}\left({x}+{y}\right)+{f}\left({x}−{y}\right)=\mathrm{2}{f}\left({x}\right){f}\left({y}\right)\wedge{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=−\mathrm{1} \\ $$$${compute}\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{sin}\:\left({k}\right)\mathrm{sin}\:\left({k}+{f}\left({k}\right)\right)}\right] \\ $$

Question Number 195592 Answers: 0 Comments: 2

f(x)=((1376)/((x−1)^(ln((2/(4689)))) )) dom f(x)=? answer this

$${f}\left({x}\right)=\frac{\mathrm{1376}}{\left({x}−\mathrm{1}\right)^{{ln}\left(\frac{\mathrm{2}}{\mathrm{4689}}\right)} } \\ $$$${dom}\:{f}\left({x}\right)=? \\ $$$${answer}\:{this} \\ $$

Question Number 199608 Answers: 2 Comments: 0

1) 3<∣2x−1∣<7 find Σx ;x∈Z 2) 4≤∣x−2∣<5 find Σx ;x∈Z

$$\left.\mathrm{1}\right)\:\:\:\mathrm{3}<\mid\mathrm{2}{x}−\mathrm{1}\mid<\mathrm{7}\:\:{find}\:\Sigma{x}\:\:\:\:;{x}\in{Z} \\ $$$$\left.\mathrm{2}\right)\:\:\:\mathrm{4}\leqslant\mid{x}−\mathrm{2}\mid<\mathrm{5}\:\:{find}\:\Sigma{x}\:\:\:\:;{x}\in{Z} \\ $$$$ \\ $$

Question Number 195511 Answers: 1 Comments: 0

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