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Question Number 158325 Answers: 2 Comments: 0

Question Number 158322 Answers: 0 Comments: 0

prove that: Σ_(n=1) ^∞ tan^( −1) ( (1/F_( n) ) ).tan^( −1) ( (1/F_( n+1) ) )= (π^( 2) /8) Fibonacci numbers

$$ \\ $$$$\:\:{prove}\:\:{that}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:{tan}^{\:−\mathrm{1}} \:\left(\:\frac{\mathrm{1}}{\mathrm{F}_{\:{n}} }\:\right).{tan}^{\:−\mathrm{1}} \left(\:\frac{\mathrm{1}}{\mathrm{F}_{\:{n}+\mathrm{1}} }\:\right)=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{8}} \\ $$$$\:\:\mathrm{F}{ibonacci}\:{numbers} \\ $$$$ \\ $$

Question Number 158320 Answers: 0 Comments: 0

prove that : Σ_(n=1) ^∞ (( H_( n) . F_n )/2^( n) ) = ln(4) + ((12)/( (√5))) ln( ϕ ) ϕ : Golden ratio F_( n) : fibonacci numbers

$$ \\ $$$$\:\:\:{prove}\:\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{H}_{\:{n}} .\:\mathrm{F}_{{n}} }{\mathrm{2}^{\:{n}} }\:\:=\:{ln}\left(\mathrm{4}\right)\:+\:\frac{\mathrm{12}}{\:\sqrt{\mathrm{5}}}\:{ln}\left(\:\varphi\:\right) \\ $$$$\:\:\:\:\:\varphi\::\:\:\:\mathrm{Golden}\:\:\mathrm{ratio} \\ $$$$\:\:\:\:\:\:\mathrm{F}_{\:{n}} \::\:{fibonacci}\:{numbers} \\ $$$$ \\ $$

Question Number 158721 Answers: 1 Comments: 0

Find: ∫_( 0) ^( ∞) (1/((x^2 + x + 1)(1 + ax))) dx ; a>0

$$\mathrm{Find}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{1}\:+\:\mathrm{ax}\right)}\:\mathrm{dx}\:\:;\:\:\mathrm{a}>\mathrm{0} \\ $$$$ \\ $$

Question Number 158313 Answers: 1 Comments: 0

Question Number 158309 Answers: 3 Comments: 0

y′′+y′−2y=−18te^(−2t)

$${y}''+{y}'−\mathrm{2}{y}=−\mathrm{18}{te}^{−\mathrm{2}{t}} \\ $$

Question Number 158306 Answers: 0 Comments: 3

Question Number 158366 Answers: 1 Comments: 0

Question Number 158303 Answers: 1 Comments: 0

x;y;z;t>0 solve for real numbers: { ((8x^4 + 64y^4 + 216z^4 + 1728t^4 = 1)),((x + y + z + t = 1)) :}

$$\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0} \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{8x}^{\mathrm{4}} \:+\:\mathrm{64y}^{\mathrm{4}} \:+\:\mathrm{216z}^{\mathrm{4}} \:+\:\mathrm{1728t}^{\mathrm{4}} \:=\:\mathrm{1}}\\{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:+\:\mathrm{t}\:=\:\mathrm{1}}\end{cases} \\ $$$$ \\ $$

Question Number 158425 Answers: 0 Comments: 0

Question Number 158424 Answers: 1 Comments: 0

Question Number 158301 Answers: 1 Comments: 1

proven that 1^0 =1 et que 0!=1

$${proven}\:{that}\: \\ $$$$\mathrm{1}^{\mathrm{0}} =\mathrm{1}\:{et}\:{que}\:\mathrm{0}!=\mathrm{1} \\ $$

Question Number 158295 Answers: 0 Comments: 0

𝛀 =∫_( 0) ^( 1) ((sin^(-1) x log(1 + x))/x^2 ) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}\:\mathrm{log}\left(\mathrm{1}\:+\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$$$ \\ $$

Question Number 158285 Answers: 0 Comments: 1

Question Number 158293 Answers: 3 Comments: 0

question# If , Ω =∫_0 ^( 1) ((ln^( 2) (1−x^( 4) ))/x) dx= a ζ b) find the value of , a , b .

$$ \\ $$$$\:\:\:\:\:{question}# \\ $$$$\left.\mathrm{If}\:,\:\:\Omega\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{\:\mathrm{4}} \right)}{{x}}\:{dx}=\:{a}\:\zeta\:{b}\right) \\ $$$$\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}\:,\:\:\:\:\:{a}\:\:,\:{b}\:\:. \\ $$$$ \\ $$$$ \\ $$

Question Number 158317 Answers: 0 Comments: 0

Question Number 158290 Answers: 0 Comments: 0

Question Number 158288 Answers: 1 Comments: 0

Question Number 158287 Answers: 0 Comments: 0

Question Number 158281 Answers: 0 Comments: 2

Question Number 158276 Answers: 0 Comments: 0

if x;y;z≥0 then: 2 Σ_(cyc) x^2 (x^2 + y^2 ) ≥ Σ_(cyc) x(x^3 + z^3 ) + xyz(x+y+z)

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}\geqslant\mathrm{0}\:\:\mathrm{then}: \\ $$$$\mathrm{2}\:\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \right)\:\geqslant\:\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\mathrm{x}\left(\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \right)\:+\:\mathrm{xyz}\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right) \\ $$$$ \\ $$

Question Number 158275 Answers: 2 Comments: 0

Solve for real numbers: x^(32) + x^(16) + y^2 = 2 (√2) x^(12) y

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{x}^{\mathrm{32}} \:+\:\mathrm{x}^{\mathrm{16}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{2}\:\sqrt{\mathrm{2}}\:\mathrm{x}^{\mathrm{12}} \:\mathrm{y} \\ $$$$ \\ $$

Question Number 158273 Answers: 0 Comments: 0

Question Number 158272 Answers: 1 Comments: 0

Question Number 158271 Answers: 0 Comments: 0

Question Number 158270 Answers: 0 Comments: 0

82,1336,18670,240004,2933338,34666672,400000006,? is there a valid pattern for these numbers?

$$\mathrm{82},\mathrm{1336},\mathrm{18670},\mathrm{240004},\mathrm{2933338},\mathrm{34666672},\mathrm{400000006},? \\ $$$$\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{valid}\:\mathrm{pattern}\:\mathrm{for}\:\mathrm{these}\:\mathrm{numbers}? \\ $$

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