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Question Number 139826    Answers: 2   Comments: 0

Σ_(n=0) ^∞ ((sin [(n−1)x])/4^(n+1) )=?

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{sin}\:\left[\left({n}−\mathrm{1}\right){x}\right]}{\mathrm{4}^{{n}+\mathrm{1}} }=? \\ $$

Question Number 139825    Answers: 0   Comments: 1

Question Number 139824    Answers: 1   Comments: 0

Solve in R the following equation: 2∙3^x +5∙4^x =4∙5^x +3∙2^x

$${Solve}\:{in}\:\mathbb{R}\:{the}\:{following}\:{equation}: \\ $$$$\mathrm{2}\centerdot\mathrm{3}^{{x}} +\mathrm{5}\centerdot\mathrm{4}^{{x}} =\mathrm{4}\centerdot\mathrm{5}^{{x}} +\mathrm{3}\centerdot\mathrm{2}^{{x}} \\ $$

Question Number 139818    Answers: 2   Comments: 0

x^4 −22x^2 +x−114=0 x=?

$${x}^{\mathrm{4}} −\mathrm{22}{x}^{\mathrm{2}} +{x}−\mathrm{114}=\mathrm{0} \\ $$$${x}=? \\ $$

Question Number 139815    Answers: 0   Comments: 3

Question Number 139814    Answers: 2   Comments: 0

Question Number 139811    Answers: 2   Comments: 0

𝛗:=∫_0 ^( 1) (√x) ln((1/(1−x)))dx solution: 𝛗:= ∫_0 ^( 1) (√x) Σ_(n=1) ^∞ (x^n /n) dx :=Σ_(n=1) ^∞ (1/n)∫_0 ^( 1) x^(n+(1/2)) dx := Σ_(n=1) ^∞ (1/(n(n+(3/2)))) = (2/3)Σ_(n=1) ^∞ ((1/n)−(1/(n+(3/2)))) :=(2/3){γ −γ+Σ_(n=1) ^∞ ((1/n)−(1/(n+(3/2)))) } := (2/3) γ +(2/3) ψ(1+(3/2))=(2/3) γ +(2/3)((2/3)+ψ((3/2))) :=(2/3) γ +(4/9) +(2/3)(2+ψ((1/2))) :=(2/3) γ +((16)/9) −(2/3) γ−(4/3) ln(2) :=((16)/9) −(4/3) ln(2) .....✓

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{{x}}\:{ln}\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right){dx} \\ $$$$\:\:\:\:\:{solution}: \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{{x}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{{n}}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\::=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}+\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{{n}+\frac{\mathrm{3}}{\mathrm{2}}}\right) \\ $$$$\:\:\:\:\:\:\::=\frac{\mathrm{2}}{\mathrm{3}}\left\{\gamma\:−\gamma+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{{n}+\frac{\mathrm{3}}{\mathrm{2}}}\right)\:\right\} \\ $$$$\:\:\:\:\::=\:\frac{\mathrm{2}}{\mathrm{3}}\:\gamma\:+\frac{\mathrm{2}}{\mathrm{3}}\:\psi\left(\mathrm{1}+\frac{\mathrm{3}}{\mathrm{2}}\right)=\frac{\mathrm{2}}{\mathrm{3}}\:\gamma\:+\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{2}}{\mathrm{3}}+\psi\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\right) \\ $$$$\:\:\:\:\::=\frac{\mathrm{2}}{\mathrm{3}}\:\gamma\:+\frac{\mathrm{4}}{\mathrm{9}}\:+\frac{\mathrm{2}}{\mathrm{3}}\left(\mathrm{2}+\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\right) \\ $$$$\:\:\:\:\::=\frac{\mathrm{2}}{\mathrm{3}}\:\gamma\:+\frac{\mathrm{16}}{\mathrm{9}}\:−\frac{\mathrm{2}}{\mathrm{3}}\:\gamma−\frac{\mathrm{4}}{\mathrm{3}}\:{ln}\left(\mathrm{2}\right)\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\::=\frac{\mathrm{16}}{\mathrm{9}}\:−\frac{\mathrm{4}}{\mathrm{3}}\:{ln}\left(\mathrm{2}\right)\:.....\checkmark\:\: \\ $$

Question Number 139800    Answers: 2   Comments: 0

(1/3)+(6/(21))+((11)/(147))+((16)/(1029))+((21)/(7203))+((26)/(50421))+…=?

$$\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{6}}{\mathrm{21}}+\frac{\mathrm{11}}{\mathrm{147}}+\frac{\mathrm{16}}{\mathrm{1029}}+\frac{\mathrm{21}}{\mathrm{7203}}+\frac{\mathrm{26}}{\mathrm{50421}}+\ldots=? \\ $$

Question Number 139799    Answers: 1   Comments: 1

x+(√y)=11 (√x)+y=7 x=? y=?

$${x}+\sqrt{{y}}=\mathrm{11} \\ $$$$\sqrt{{x}}+{y}=\mathrm{7} \\ $$$${x}=?\:\:\:\:\:\:\:{y}=? \\ $$

Question Number 139789    Answers: 2   Comments: 0

......Mathematical ... ... ... Analysis....... evaluation :: F :=∫_0 ^( ∞) e^((−2)/x) sin^2 ((2/x))dx=?

$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:......\mathscr{M}{athematical}\:...\:...\:...\:\mathscr{A}{nalysis}....... \\ $$$$\:\:\:\:{evaluation}\:::\:\mathscr{F}\::=\int_{\mathrm{0}} ^{\:\infty} {e}^{\frac{−\mathrm{2}}{{x}}} {sin}^{\mathrm{2}} \left(\frac{\mathrm{2}}{{x}}\right){dx}=? \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 139776    Answers: 3   Comments: 0

∫ (dx/(2cos(x)+3sin(x)))

$$\int\:\frac{{dx}}{\mathrm{2}{cos}\left({x}\right)+\mathrm{3}{sin}\left({x}\right)} \\ $$

Question Number 139778    Answers: 2   Comments: 0

prove that the absolute valje of z1+z2<=absolute value of z1+absolute value of z2

$${prove}\:{that}\:{the}\:{absolute}\:{valje}\:{of}\:{z}\mathrm{1}+{z}\mathrm{2}<={absolute}\:{value}\:{of}\:{z}\mathrm{1}+{absolute}\:{value}\:{of}\:{z}\mathrm{2} \\ $$

Question Number 139805    Answers: 1   Comments: 1

solve for x∈R x^x^n =a with n, a∈R^+ find also the range of a such that a solution exists.

$${solve}\:{for}\:{x}\in\mathbb{R} \\ $$$$\boldsymbol{{x}}^{\boldsymbol{{x}}^{\boldsymbol{{n}}} } =\boldsymbol{{a}} \\ $$$${with}\:{n},\:{a}\in\mathbb{R}^{+} \\ $$$${find}\:{also}\:{the}\:{range}\:{of}\:\boldsymbol{{a}}\:{such}\:{that}\:{a} \\ $$$${solution}\:{exists}. \\ $$

Question Number 139771    Answers: 3   Comments: 5

Question Number 139768    Answers: 1   Comments: 2

x+(√y)=11 (√x)+(√y)=7 x=? y=?

$${x}+\sqrt{{y}}=\mathrm{11} \\ $$$$\sqrt{{x}}+\sqrt{{y}}=\mathrm{7} \\ $$$${x}=? \\ $$$${y}=? \\ $$

Question Number 139766    Answers: 1   Comments: 0

lim_(x→0) (((√(1−cos(2x)))/(1−cos(x))))

$${lim}_{{x}\rightarrow\mathrm{0}} \left(\frac{\sqrt{\mathrm{1}−{cos}\left(\mathrm{2}{x}\right)}}{\mathrm{1}−{cos}\left({x}\right)}\right) \\ $$

Question Number 139765    Answers: 0   Comments: 1

Question Number 139762    Answers: 1   Comments: 0

x;y;z>0, γ≥0, x^3 +y^3 +z^3 +xyz=4 proof: (x+y+z)^3 +γ(x^3 +y^3 +z^3 )≥27+3γ

$${x};{y};{z}>\mathrm{0},\:\gamma\geqslant\mathrm{0},\:{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} +{xyz}=\mathrm{4} \\ $$$${proof}:\:\left({x}+{y}+{z}\right)^{\mathrm{3}} +\gamma\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} \right)\geqslant\mathrm{27}+\mathrm{3}\gamma \\ $$

Question Number 139756    Answers: 1   Comments: 0

Question Number 139750    Answers: 0   Comments: 0

∫_0 ^1 ((1−x^7 )^(1/3) (1−x^3 )^(1/7) )dx =?

$$\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\left(\mathrm{1}−\mathrm{x}^{\mathrm{7}} \right)^{\mathrm{1}/\mathrm{3}} \:\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{7}} \:\right)\mathrm{dx}\:=? \\ $$

Question Number 139746    Answers: 1   Comments: 0

Question Number 139745    Answers: 1   Comments: 0

n ∈ N. a_n =5^(n+1) +5^n We can say that: • a_n is always odd. • a_n is always even. • It depends of n. • The sign of a_n depends of n.

$${n}\:\in\:\mathbb{N}. \\ $$$${a}_{{n}} =\mathrm{5}^{{n}+\mathrm{1}} +\mathrm{5}^{{n}} \:{We}\:{can}\:{say}\:{that}: \\ $$$$\bullet\:{a}_{{n}} \:{is}\:{always}\:{odd}. \\ $$$$\bullet\:{a}_{{n}} \:{is}\:{always}\:{even}. \\ $$$$\bullet\:{It}\:{depends}\:{of}\:{n}. \\ $$$$\bullet\:{The}\:{sign}\:{of}\:{a}_{{n}} \:{depends}\:{of}\:{n}. \\ $$$$ \\ $$$$ \\ $$

Question Number 139744    Answers: 0   Comments: 1

log_2 (5^2 ×2^4 )=?

$${log}_{\mathrm{2}} \left(\mathrm{5}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{4}} \right)=? \\ $$

Question Number 139743    Answers: 0   Comments: 0

Please how do we calculate the area of an hexagon?

$${Please}\:{how}\:{do}\:{we}\:{calculate}\:{the} \\ $$$${area}\:{of}\:{an}\:{hexagon}? \\ $$

Question Number 139742    Answers: 1   Comments: 0

Today is the birthday of Matongue. His age is the double of his daughter′s age. Will this event happen in the futur? a.Yes Each two years b. Yes In two years. c. No d. Any correct answer.

$${Today}\:{is}\:{the}\:{birthday}\:{of}\:{Matongue}. \\ $$$${His}\:{age}\:{is}\:{the}\:{double}\:{of}\:{his}\:{daughter}'{s} \\ $$$${age}.\:{Will}\:{this}\:{event}\:{happen}\:{in}\:{the} \\ $$$${futur}? \\ $$$${a}.{Yes}\:{Each}\:{two}\:{years} \\ $$$${b}.\:{Yes}\:{In}\:{two}\:{years}. \\ $$$${c}.\:{No} \\ $$$${d}.\:{Any}\:{correct}\:{answer}. \\ $$

Question Number 139741    Answers: 2   Comments: 0

A man invite 5 friends choosen from 10 for a diner. He don′t want that two of them took part to that diner. The number of ways to choose her friends is: a)296 b)196 c)5 d)252

$${A}\:{man}\:{invite}\:\mathrm{5}\:{friends}\:{choosen} \\ $$$${from}\:\mathrm{10}\:{for}\:{a}\:{diner}.\:{He}\:{don}'{t}\: \\ $$$${want}\:{that}\:{two}\:{of}\:{them}\:{took}\:{part} \\ $$$${to}\:{that}\:{diner}.\:{The}\:{number}\:{of} \\ $$$${ways}\:{to}\:{choose}\:{her}\:{friends}\:{is}: \\ $$$$\left.{a}\right)\mathrm{296} \\ $$$$\left.{b}\right)\mathrm{196} \\ $$$$\left.{c}\right)\mathrm{5} \\ $$$$\left.{d}\right)\mathrm{252} \\ $$

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