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AllQuestion and Answers: Page 587

Question Number 153915    Answers: 1   Comments: 0

Question Number 153912    Answers: 2   Comments: 0

Question Number 153905    Answers: 2   Comments: 1

Question Number 153903    Answers: 0   Comments: 0

Question Number 153901    Answers: 2   Comments: 1

Question Number 153899    Answers: 0   Comments: 0

Determine whether there exists 2016 distinct prime numbers p_1 ,p_2 ,...,p_(2016) and positive integer n such that: Σ_(i=1) ^(2016) (1/(p_i ^2 + 1)) = (1/n^2 )

$$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{there}\:\mathrm{exists}\:\:\mathrm{2016} \\ $$$$\mathrm{distinct}\:\mathrm{prime}\:\mathrm{numbers}\:\:\mathrm{p}_{\mathrm{1}} ,\mathrm{p}_{\mathrm{2}} ,...,\mathrm{p}_{\mathrm{2016}} \\ $$$$\mathrm{and}\:\mathrm{positive}\:\mathrm{integer}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\mathrm{2016}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{p}_{\boldsymbol{\mathrm{i}}} ^{\mathrm{2}} \:+\:\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 153898    Answers: 1   Comments: 0

Find all functions f:Q→Q satisfying these followong conditions for all x∈Q 1. f(x + 1) = f(x) + 1 2. f(x^3 ) = f^( 3) (x)

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{functions}\:\:\mathrm{f}:\mathrm{Q}\rightarrow\mathrm{Q}\:\:\mathrm{satisfying} \\ $$$$\mathrm{these}\:\mathrm{followong}\:\mathrm{conditions}\:\mathrm{for}\:\mathrm{all}\:\boldsymbol{\mathrm{x}}\in\mathrm{Q} \\ $$$$\mathrm{1}.\:\mathrm{f}\left(\mathrm{x}\:+\:\mathrm{1}\right)\:=\:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{1} \\ $$$$\mathrm{2}.\:\mathrm{f}\left(\mathrm{x}^{\mathrm{3}} \right)\:=\:\mathrm{f}^{\:\mathrm{3}} \left(\mathrm{x}\right) \\ $$

Question Number 153896    Answers: 1   Comments: 0

Question Number 153895    Answers: 0   Comments: 0

Question Number 153897    Answers: 0   Comments: 1

Denote x_n is the unique positive root of the following equation: x^n + x^(n−1) + ... x = n + 2 Prove that the sequence (x_n ) converges to a positive real number. Find that limit.

$$\mathrm{Denote}\:\:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \:\:\mathrm{is}\:\mathrm{the}\:\mathrm{unique}\:\mathrm{positive}\:\mathrm{root} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{x}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \:+\:...\:\mathrm{x}\:=\:\mathrm{n}\:+\:\mathrm{2} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left(\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)\:\mathrm{converges} \\ $$$$\mathrm{to}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{Find}\:\mathrm{that} \\ $$$$\mathrm{limit}. \\ $$

Question Number 153893    Answers: 0   Comments: 0

∫_( 0) ^( ∞) a Π_(p → 1) ^∞ (((p^2 − x^(2n) )/p^2 ))dx, 1 < 2n < n + 1

$$\int_{\:\mathrm{0}} ^{\:\:\infty} \mathrm{a}\:\underset{\mathrm{p}\:\rightarrow\:\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{p}^{\mathrm{2}} \:\:−\:\:\:\mathrm{x}^{\mathrm{2n}} }{\mathrm{p}^{\mathrm{2}} }\right)\mathrm{dx},\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:<\:\:\mathrm{2n}\:\:<\:\:\mathrm{n}\:\:+\:\:\mathrm{1} \\ $$

Question Number 153877    Answers: 0   Comments: 1

Question Number 155421    Answers: 3   Comments: 0

Question Number 153875    Answers: 0   Comments: 0

Prove that.. 𝛗 : =∫_( 1) ^( +∞) (( ln (x ))/(( x^( π) −1 )( ln^( 2) (x) +1 )^2 ))dx= ((π^( 2) − 8)/(16)) ■

$$ \\ $$$$\:\:\:\:\mathrm{Prove}\:\:\mathrm{that}.. \\ $$$$\:\:\: \\ $$$$\:\:\:\:\boldsymbol{\phi}\::\:=\int_{\:\mathrm{1}} ^{\:+\infty} \frac{\:{ln}\:\left({x}\:\right)}{\left(\:{x}^{\:\pi} \:−\mathrm{1}\:\right)\left(\:{ln}^{\:\mathrm{2}} \left({x}\right)\:+\mathrm{1}\:\right)^{\mathrm{2}} }{dx}=\:\frac{\pi^{\:\mathrm{2}} −\:\mathrm{8}}{\mathrm{16}}\:\:\:\:\:\:\:\:\:\blacksquare\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 153873    Answers: 0   Comments: 1

∫_0 ^( ∞) (( x)/((1 +x^( 2) ) ( e^( 2πx) − 1))) dx =((2γ− 1)/4)

$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}}{\left(\mathrm{1}\:+{x}^{\:\mathrm{2}} \right)\:\left(\:{e}^{\:\mathrm{2}\pi{x}} −\:\mathrm{1}\right)}\:{dx}\:=\frac{\mathrm{2}\gamma−\:\mathrm{1}}{\mathrm{4}} \\ $$$$ \\ $$

Question Number 153870    Answers: 1   Comments: 3

find the minimum and maximum value of (5/(f(θ)+3)) where f(θ)=8cos θ−15 sin θ

$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{5}}{{f}\left(\theta\right)+\mathrm{3}}\:\mathrm{where}\:{f}\left(\theta\right)=\mathrm{8cos}\:\theta−\mathrm{15}\:\mathrm{sin}\:\theta \\ $$

Question Number 153864    Answers: 0   Comments: 1

Question Number 153866    Answers: 1   Comments: 0

3+(√(3+(√(6+(√(9+(√(12+…+(√(99))))))))))=?

$$\mathrm{3}+\sqrt{\mathrm{3}+\sqrt{\mathrm{6}+\sqrt{\mathrm{9}+\sqrt{\mathrm{12}+\ldots+\sqrt{\mathrm{99}}}}}}=? \\ $$

Question Number 153862    Answers: 0   Comments: 0

Question Number 153860    Answers: 1   Comments: 0

Question Number 153858    Answers: 2   Comments: 1

Question Number 153857    Answers: 1   Comments: 1

Question Number 153847    Answers: 2   Comments: 0

S = x + 2x^2 + ... + nx^n

$$\boldsymbol{\mathrm{S}}\:=\:\mathrm{x}\:+\:\mathrm{2x}^{\mathrm{2}} \:+\:...\:+\:\mathrm{nx}^{\boldsymbol{\mathrm{n}}} \\ $$$$ \\ $$

Question Number 153840    Answers: 1   Comments: 0

log _e (x)+log _x (e)+log _(((e/x))) (x)=(5/2) x=?

$$\:\:\:\:\mathrm{log}\:_{{e}} \left({x}\right)+\mathrm{log}\:_{{x}} \left({e}\right)+\mathrm{log}\:_{\left(\frac{{e}}{{x}}\right)} \left({x}\right)=\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$\:{x}=? \\ $$

Question Number 153839    Answers: 1   Comments: 1

Question Number 153829    Answers: 0   Comments: 0

L(x) = (6/π^2 )(Li_2 (x) + (1/2) log(x)log(1-x) Find: 𝛀 =∫_( 0) ^( 1) L(x)∙Li_2 (x) dx

$$\mathrm{L}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{6}}{\pi^{\mathrm{2}} }\left(\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{log}\left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{1}-\mathrm{x}\right)\right. \\ $$$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{L}\left(\mathrm{x}\right)\centerdot\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

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