Question and Answers Forum

1) prove the convexity of f(x)=ln(1+e^x ) 2) prove that ∀(x_1 ,x_2 ,...,x_n )∈R^n 1+Π_(k=1) ^n (x_k )^(1/n) ≤ Π_(k=1) ^n (x_k +1)^(1/n) 3) prove that 1+(n!)^(1/n) ≤((n+1)!)^(1/n)

$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{convexity}\:{of}\:{f}\left({x}\right)={ln}\left(\mathrm{1}+{e}^{{x}} \right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...,{x}_{{n}} \right)\in{R}^{{n}} \\ $$$$\mathrm{1}+\prod_{{k}=\mathrm{1}} ^{{n}} \left({x}_{{k}} \right)^{\frac{\mathrm{1}}{{n}}} \:\leqslant\:\prod_{{k}=\mathrm{1}} ^{{n}} \left({x}_{{k}} +\mathrm{1}\right)^{\frac{\mathrm{1}}{{n}}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\mathrm{1}+\left({n}!\right)^{\frac{\mathrm{1}}{{n}}} \leqslant\left(\left({n}+\mathrm{1}\right)!\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 51983 Answers: 0 Comments: 0

f is a real function derivable at 0 and f(0)=0 find lim_(n→+∞) Σ_(k=1) ^n f((k/n^2 )) .

$${f}\:{is}\:{a}\:{real}\:{function}\:{derivable}\:{at}\:\mathrm{0}\:{and}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{find} \\ $$$${lim}_{{n}\rightarrow+\infty} \sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:. \\ $$

Question Number 51982 Answers: 0 Comments: 0

let u_(n+1) =(√(Σ_(k=1) ^n u_k )) with n>0 and u_1 =1 1)calculate u_2 ,u_3 ,u_4 and u_5 2)prove that ∀n≥2 u_(n+) ^2 =u_n ^2 +u_n 3)study the variation of u_n 4)prove that lim_(n→+∞) u_n =+∞ 5)prove that u_(n+1) ∼u_n (n→+∞) 6)let v_n =u_(n+1) −u_n prove that (v_n ) converges and find its limit.

$${let}\:{u}_{{n}+\mathrm{1}} =\sqrt{\sum_{{k}=\mathrm{1}} ^{{n}} \:{u}_{{k}} }\:\:\:\:\:\:\:{with}\:{n}>\mathrm{0}\:\:\:{and}\:{u}_{\mathrm{1}} =\mathrm{1} \\ $$$$\left.\mathrm{1}\right){calculate}\:{u}_{\mathrm{2}} ,{u}_{\mathrm{3}} ,{u}_{\mathrm{4}} {and}\:{u}_{\mathrm{5}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:\forall{n}\geqslant\mathrm{2}\:\:\:\:\:{u}_{{n}+} ^{\mathrm{2}} ={u}_{{n}} ^{\mathrm{2}} \:+{u}_{{n}} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{variation}\:{of}\:{u}_{{n}} \\ $$$$\left.\mathrm{4}\right){prove}\:{that}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} =+\infty \\ $$$$\left.\mathrm{5}\right){prove}\:{that}\:{u}_{{n}+\mathrm{1}} \sim{u}_{{n}} \:\:\left({n}\rightarrow+\infty\right) \\ $$$$\left.\mathrm{6}\right){let}\:{v}_{{n}} ={u}_{{n}+\mathrm{1}} −{u}_{{n}} \:\:{prove}\:{that}\:\left({v}_{{n}} \right)\:{converges}\:{and}\:{find}\:{its}\:{limit}. \\ $$

Question Number 51981 Answers: 0 Comments: 0

let U_n = ((Σ_(k=1) ^n [lnk])/(ln(n!))) determine lim_(n→+∞) U_n

$${let}\:\:{U}_{{n}} =\:\frac{\sum_{{k}=\mathrm{1}} ^{{n}} \left[{lnk}\right]}{{ln}\left({n}!\right)}\:\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 51980 Answers: 0 Comments: 0

let S_n =Σ_(k=1) ^(2n+1) (1/(√(n^2 +k))) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{\mathrm{2}{n}+\mathrm{1}} \:\:\:\:\frac{\mathrm{1}}{\sqrt{{n}^{\mathrm{2}} +{k}}}\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} \\ $$

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Question Number 51950 Answers: 2 Comments: 0

a^2 + b^2 + c^2 = 2019 a, b, c are prime numbers . how many possible triples of (a, b, c) which that suitable for equation above .

$${a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:\:=\:\:\mathrm{2019} \\ $$$${a},\:{b},\:{c}\:\:\:{are}\:\:{prime}\:\:{numbers}\:. \\ $$$${how}\:\:{many}\:\:{possible}\:\:{triples}\:\:{of}\:\:\left({a},\:{b},\:{c}\right)\:\:{which}\:\:{that}\:\:{suitable}\:\:{for}\:\:{equation}\:\:{above}\:. \\ $$

Question Number 51942 Answers: 1 Comments: 5

Question Number 51938 Answers: 0 Comments: 2

To which interaction among the four fundamental interactions is linked the combustion ? the fission ? Thanks.

$$\mathrm{To}\:\mathrm{which}\:\mathrm{interaction}\:\mathrm{among}\:\mathrm{the}\:\mathrm{four} \\ $$$$\mathrm{fundamental}\:\mathrm{interactions}\:\mathrm{is}\:\mathrm{linked} \\ $$$$\mathrm{the}\:\mathrm{combustion}\:?\:\mathrm{the}\:\mathrm{fission}\:? \\ $$$$ \\ $$$$\mathrm{Thanks}. \\ $$

Question Number 51933 Answers: 2 Comments: 1

If p = cos θ + i sin θ and q = cos φ + i sin φ Show that (((p + q)(pq − 1))/((p − q)(pq + 1))) = ((sin θ + sin φ)/(sin θ − sin φ))

$$\mathrm{If}\:\:\mathrm{p}\:=\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\:\theta\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$

Question Number 51922 Answers: 1 Comments: 0

find the value of... 1−(1/(1+(1/(i/(1+(i/(1+i))))))) pls help.

$${find}\:{the}\:{value}\:{of}... \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\frac{{i}}{\mathrm{1}+\frac{{i}}{\mathrm{1}+{i}}}}} \\ $$$${pls}\:{help}. \\ $$

Question Number 51921 Answers: 1 Comments: 0

A normal chord to an ellipse (x^2 /a^2 )+(y^2 /b^2 )=1 make an angle of 45^° with the axis.prove that the square of its length is equal to ((32a^4 b^4 )/((a^2 +b^2 )^3 ))

$${A}\:{normal}\:{chord}\:{to}\:{an}\: \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$${make}\:{an}\:{angle}\:{of}\:\mathrm{45}^{°} \\ $$$${with}\:{the}\:{axis}.{prove} \\ $$$${that}\:{the}\:{square}\:{of}\:{its}\: \\ $$$${length}\:{is}\:{equal}\:{to} \\ $$$$\frac{\mathrm{32}{a}^{\mathrm{4}} {b}^{\mathrm{4}} }{\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{3}} }\: \\ $$

Question Number 51911 Answers: 0 Comments: 2

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Question Number 51905 Answers: 2 Comments: 0

If p = cos θ + i sinθ and q = cos φ + i sin φ Show that: (i) ((p − q)/(p + q)) = i tan (((θ − φ)/2)) (ii) (((p + q)(pq − 1))/((p − q)(pq + 1))) = ((sin θ + sin φ)/(sin θ − sin φ))

$$\mathrm{If}\:\:\:\mathrm{p}\:\:=\:\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\theta\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}: \\ $$$$\left(\mathrm{i}\right)\:\:\:\:\:\:\frac{\mathrm{p}\:−\:\mathrm{q}}{\mathrm{p}\:+\:\mathrm{q}}\:\:=\:\:\mathrm{i}\:\mathrm{tan}\:\left(\frac{\theta\:−\:\phi}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{ii}\right)\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$

Question Number 51901 Answers: 0 Comments: 0

Prove that the two parabola y^2 =4ax and y^2 =4c(x−b) cannot have a common normal other than the axis unless (b/(a−c))>2

$${Prove}\:{that}\:{the}\:{two}\: \\ $$$${parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:{and} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{c}\left({x}−{b}\right)\:{cannot} \\ $$$${have}\:{a}\:{common}\:{normal} \\ $$$${other}\:{than}\:{the}\:{axis}\:{unless} \\ $$$$\frac{{b}}{{a}−{c}}>\mathrm{2} \\ $$

Question Number 51897 Answers: 1 Comments: 0

If x + (1/x) = 2cosθ , y + (1/y) = 2cosφ , z + (1/z) = 2cosψ Show that xyz + (1/(xyz)) = 2cos(θ + φ + ψ)

$$\mathrm{If}\:\:\:\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\:\:=\:\:\mathrm{2cos}\theta\:,\:\:\:\:\:\:\mathrm{y}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:\:=\:\:\mathrm{2cos}\phi\:,\:\:\:\:\:\:\:\:\:\mathrm{z}\:+\:\frac{\mathrm{1}}{\mathrm{z}}\:\:=\:\:\mathrm{2cos}\psi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\mathrm{xyz}\:+\:\frac{\mathrm{1}}{\mathrm{xyz}}\:\:=\:\:\mathrm{2cos}\left(\theta\:+\:\phi\:+\:\psi\right) \\ $$

Question Number 51887 Answers: 1 Comments: 0

Prove that; tanh(log (√3)) = (1/2)

$$\mathrm{Prove}\:\mathrm{that};\:\:\:\:\mathrm{tanh}\left(\mathrm{log}\:\sqrt{\mathrm{3}}\right)\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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