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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

Question Number 150392    Answers: 0   Comments: 0

Question Number 150376    Answers: 1   Comments: 2

Question Number 150374    Answers: 1   Comments: 0

Σ_(k=0) ^∞ ((2^k + 3^k )/5^k ) = ?

$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{5}^{\boldsymbol{\mathrm{k}}} }\:\:=\:? \\ $$

Question Number 150372    Answers: 0   Comments: 1

Question Number 150450    Answers: 0   Comments: 0

show the?connection between the beta distribution(n,p) and hypergeometric distribution(N,k,n)in a limiting case

$$\mathrm{show}\:\mathrm{the}?\mathrm{connection}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{beta}\:\mathrm{distribution}\left(\mathrm{n},\mathrm{p}\right)\:\mathrm{and}\:\mathrm{hypergeometric} \\ $$$$\mathrm{distribution}\left(\mathrm{N},\mathrm{k},\mathrm{n}\right)\mathrm{in}\:\mathrm{a}\:\mathrm{limiting}\:\mathrm{case} \\ $$

Question Number 150366    Answers: 0   Comments: 1

Question Number 150405    Answers: 2   Comments: 0

Question Number 150404    Answers: 4   Comments: 0

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

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

Question Number 150402    Answers: 1   Comments: 1

For m≥1 𝛀 =∫_( 0) ^( ∞) ((x ln^m (x))/(e^x − 1)) = 2 ln^m 𝛇(3)

$$\mathrm{For}\:\:\mathrm{m}\geqslant\mathrm{1} \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{x}\:\mathrm{ln}^{\boldsymbol{\mathrm{m}}} \:\left(\mathrm{x}\right)}{\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{1}}\:=\:\mathrm{2}\:\mathrm{ln}^{\boldsymbol{\mathrm{m}}} \:\boldsymbol{\zeta}\left(\mathrm{3}\right) \\ $$

Question Number 150400    Answers: 0   Comments: 0

Question Number 150363    Answers: 2   Comments: 5

Question Number 150469    Answers: 1   Comments: 0

If f(x) = (4^x /(4^x + 2)) find f((1/(17))) + f(((16)/(17))) =^(?)

$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{4}^{\boldsymbol{\mathrm{x}}} }{\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}}\:\:\:\mathrm{find}\:\:\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{17}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{16}}{\mathrm{17}}\right)\:\overset{?} {=} \\ $$

Question Number 150351    Answers: 1   Comments: 2

Question Number 150350    Answers: 0   Comments: 0

Question Number 150346    Answers: 0   Comments: 2

If x;y and z are positive integers, then determine the smallest positive integer N=x+y+z+xy+yz+zx, which is bigger than 2022.

$$\mathrm{If}\:\:\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}}\:\mathrm{and}\:\boldsymbol{\mathrm{z}}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer} \\ $$$$\boldsymbol{\mathrm{N}}=\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{xy}+\mathrm{yz}+\mathrm{zx},\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{bigger}\:\mathrm{than}\:\mathrm{2022}. \\ $$

Question Number 150332    Answers: 0   Comments: 0

Question Number 150331    Answers: 0   Comments: 0

ultimately Q.149894 boils down to finding s_(max) , s_(min) ∀ {h−scos (θ−(π/6))}^2 +{k−ssin (θ+(π/6))}^2 =r^2

$${ultimately}\:\:{Q}.\mathrm{149894}\:\:{boils}\: \\ $$$${down}\:{to}\:{finding}\:{s}_{{max}} ,\:{s}_{{min}} \:\forall \\ $$$$\:\:\:\:\left\{{h}−{s}\mathrm{cos}\:\left(\theta−\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} \\ $$$$+\left\{{k}−{s}\mathrm{sin}\:\left(\theta+\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 150328    Answers: 0   Comments: 0

Find all the solutions to the equation: x^3 − y^3 = xy + 61

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{y}}^{\mathrm{3}} \:=\:\boldsymbol{\mathrm{xy}}\:+\:\mathrm{61} \\ $$

Question Number 150327    Answers: 2   Comments: 0

1) 33x ≡ 48 (mod 654) 2) 5^(1000 000) ≡ x (mod 41) Find x=?

$$\left.\mathrm{1}\right)\:\mathrm{33}\boldsymbol{\mathrm{x}}\:\:\equiv\:\:\mathrm{48}\:\left(\mathrm{mod}\:\mathrm{654}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{5}^{\mathrm{1000}\:\mathrm{000}} \:\:\equiv\:\:\boldsymbol{\mathrm{x}}\:\left(\mathrm{mod}\:\mathrm{41}\right) \\ $$$$\mathrm{Find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

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