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Question Number 150550 Answers: 0 Comments: 3

Question Number 150549 Answers: 0 Comments: 0

Prove that: ∀n∈N Π_(k=1) ^n k! ∙ k^(n−k+1) ≤ (((n+2)/3))^(n∙(n+1))

$$\mathrm{Prove}\:\mathrm{that}:\:\:\forall\mathrm{n}\in\mathbb{N} \\ $$$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\mathrm{k}!\:\centerdot\:\mathrm{k}^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}+\mathrm{1}} \:\leqslant\:\left(\frac{\mathrm{n}+\mathrm{2}}{\mathrm{3}}\right)^{\boldsymbol{\mathrm{n}}\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)} \\ $$

Question Number 150548 Answers: 0 Comments: 0

For k<N fixed and 𝛂>0 then: lim_(n→∞) (1/( (√n^𝛂 ))) ∙ (((Π_(i=1) ^k (n+k+i))/(Π_(i=1) ^k (n+i))))^n^𝛂

$$\mathrm{For}\:\:\boldsymbol{\mathrm{k}}<\mathbb{N}\:\:\mathrm{fixed}\:\:\mathrm{and}\:\:\boldsymbol{\alpha}>\mathrm{0}\:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}^{\boldsymbol{\alpha}} }}\:\centerdot\:\left(\frac{\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\prod}}\left(\mathrm{n}+\mathrm{k}+\mathrm{i}\right)}{\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\prod}}\left(\mathrm{n}+\mathrm{i}\right)}\right)^{\boldsymbol{\mathrm{n}}^{\boldsymbol{\alpha}} } \\ $$

Question Number 150540 Answers: 0 Comments: 0

...solve... Ω := Σ_(n=0) ^∞ (( n)/(e^( 2nπ) − 1)) =?

$$\:\:\:...\mathrm{solve}... \\ $$$$\:\:\:\:\:\:\:\:\Omega\::=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\frac{\:{n}}{{e}^{\:\mathrm{2}{n}\pi} \:−\:\mathrm{1}}\:=? \\ $$

Question Number 150539 Answers: 1 Comments: 0

Prove or disprove the foolowing: Σ_(n=1) ^∞ (−1)^((n^2 +n+2)/2) e^(−𝛑n^2 x) = Σ_(n=1) ^∞ e^(−𝛑n^2 x)

$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{the}\:\mathrm{foolowing}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}+\mathrm{2}}{\mathrm{2}}} \:\mathrm{e}^{−\boldsymbol{\pi\mathrm{n}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:=\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{e}^{−\boldsymbol{\pi\mathrm{n}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \\ $$

Question Number 150527 Answers: 1 Comments: 0

((x(x−3)(x−9)−8))^(1/3) =2x+((x^3 −3x))^(1/3)

$$\sqrt[{\mathrm{3}}]{\mathrm{x}\left(\mathrm{x}−\mathrm{3}\right)\left(\mathrm{x}−\mathrm{9}\right)−\mathrm{8}}=\mathrm{2x}+\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{3}} −\mathrm{3x}} \\ $$

Question Number 150517 Answers: 0 Comments: 6

Question Number 150516 Answers: 1 Comments: 2

Compare: x = 2^3^2^3 and y = 3^2^3^2

$$\mathrm{Compare}: \\ $$$$\boldsymbol{\mathrm{x}}\:=\:\mathrm{2}^{\mathrm{3}^{\mathrm{2}^{\mathrm{3}} } } \:\:\:\:\:\mathrm{and}\:\:\:\:\:\boldsymbol{\mathrm{y}}\:=\:\mathrm{3}^{\mathrm{2}^{\mathrm{3}^{\mathrm{2}} } } \\ $$

Question Number 150515 Answers: 2 Comments: 0

Question Number 150508 Answers: 1 Comments: 1

Question Number 150502 Answers: 0 Comments: 0

Question Number 150501 Answers: 0 Comments: 0

Question Number 150500 Answers: 1 Comments: 1

Find sum of first-n terms in u_n =4, 9, 15, 23, 35, 55, 91, 159,...

$$\mathrm{Find}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}-\mathrm{n}\:\mathrm{terms}\:\mathrm{in} \\ $$$${u}_{{n}} =\mathrm{4},\:\mathrm{9},\:\mathrm{15},\:\mathrm{23},\:\mathrm{35},\:\mathrm{55},\:\mathrm{91},\:\mathrm{159},... \\ $$

Question Number 150489 Answers: 1 Comments: 0

∫_( 0) ^( 2) ∣x∣ x^([x+1]) sgn(x) dx = ?

$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{2}} {\int}}\:\mid\mathrm{x}\mid\:\mathrm{x}^{\left[\boldsymbol{\mathrm{x}}+\mathrm{1}\right]} \:\mathrm{sgn}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 150486 Answers: 0 Comments: 2

x ∈ R ∣x - 1∣ + ∣x + 3∣ + ∣x - 5∣ find the smallest value of a given expression

$$\boldsymbol{\mathrm{x}}\:\in\:\mathbb{R} \\ $$$$\mid\mathrm{x}\:-\:\mathrm{1}\mid\:+\:\mid\mathrm{x}\:+\:\mathrm{3}\mid\:+\:\mid\mathrm{x}\:-\:\mathrm{5}\mid \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}\:\mathrm{given} \\ $$$$\mathrm{expression} \\ $$

Question Number 150453 Answers: 0 Comments: 0

In △ABC , △A^′ B^′ C^′ the following relationship holds: R^2 R^′ F^′ ≥ 8F(r^′ )^3

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\bigtriangleup\mathrm{A}^{'} \mathrm{B}^{'} \mathrm{C}^{'} \:\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{relationship}\:\mathrm{holds}: \\ $$$$\mathrm{R}^{\mathrm{2}} \mathrm{R}^{'} \mathrm{F}^{'} \:\geqslant\:\mathrm{8F}\left(\mathrm{r}^{'} \right)^{\mathrm{3}} \\ $$

Question Number 150451 Answers: 2 Comments: 2

Question Number 150446 Answers: 0 Comments: 0

Question Number 150435 Answers: 1 Comments: 0

Solve the equation: ∣x - 3∣^((x^2 - 8x + 15)/(x - 2)) = 1

$$\boldsymbol{\mathrm{S}}\mathrm{olve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mid\mathrm{x}\:-\:\mathrm{3}\mid^{\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:-\:\mathrm{8x}\:+\:\mathrm{15}}{\boldsymbol{\mathrm{x}}\:-\:\mathrm{2}}} \:=\:\mathrm{1} \\ $$

Question Number 150432 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (((2n+1)!)/(8^n ∙(n!)^2 ))=? Help please

$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2n}+\mathrm{1}\right)!}{\mathrm{8}^{\mathrm{n}} \centerdot\left(\mathrm{n}!\right)^{\mathrm{2}} }=?\:\:\:\:\:\mathrm{Help}\:\mathrm{please} \\ $$

Question Number 150429 Answers: 1 Comments: 0

Find the equations of the common tangents to the parabola y^2 =4x and the parabola x^2 =2y−3.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{common} \\ $$$$\mathrm{tangents}\:\mathrm{to}\:\mathrm{the}\:\mathrm{parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{parabola}\:{x}^{\mathrm{2}} =\mathrm{2}{y}−\mathrm{3}. \\ $$

Question Number 150425 Answers: 0 Comments: 6

Question Number 150421 Answers: 0 Comments: 0

Question Number 150418 Answers: 2 Comments: 3

Question Number 150413 Answers: 0 Comments: 3

Question Number 150410 Answers: 1 Comments: 1

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