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Question Number 148635    Answers: 0   Comments: 0

show that {a_n }:=(1/(1!))−(1/(2!))+(1/(3!))−(1/(4!))...+(((−1)^(n+1) )/(n!)) is a cauchy sequence. my attempt: let ε>0 we have ∣a_m −a_n ∣=∣(((−1)^(n+2) )/((n+1)!))+(((−1)^(n+3) )/((n+2)!))+...+(((−1)^(m+1) )/(m!))∣ ≤∣(((−1)^(n+2) )/((n+1)!))∣+∣(((−1)^(n+3) )/((n+2)!))∣+...+∣(((−1)^(m+1) )/(m!))∣ =(1/((n+1)!))+(1/((n+2)!))+...+(1/(m!))<(1/(n!))≤(1/n)<ε. (1/n)→0 as n→∞, so no matter any ε>0 (1/n)<ε eventually. as long as n^∗ >(1/ε) ∣a_m −a_n ∣<ε ∀n≥n^∗ .

$$\mathrm{show}\:\mathrm{that}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}:=\frac{\mathrm{1}}{\mathrm{1}!}−\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{3}!}−\frac{\mathrm{1}}{\mathrm{4}!}...+\frac{\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{1}} }{\mathrm{n}!}\:\mathrm{is}\:\mathrm{a}\:\mathrm{cauchy} \\ $$$$\mathrm{sequence}. \\ $$$$\mathrm{my}\:\mathrm{attempt}: \\ $$$$\mathrm{let}\:\epsilon>\mathrm{0}\:\mathrm{we}\:\mathrm{have}\:\mid\mathrm{a}_{\mathrm{m}} −\mathrm{a}_{\mathrm{n}} \mid=\mid\frac{\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{2}} }{\left(\mathrm{n}+\mathrm{1}\right)!}+\frac{\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{3}} }{\left(\mathrm{n}+\mathrm{2}\right)!}+...+\frac{\left(−\mathrm{1}\right)^{\mathrm{m}+\mathrm{1}} }{\mathrm{m}!}\mid \\ $$$$\leqslant\mid\frac{\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{2}} }{\left(\mathrm{n}+\mathrm{1}\right)!}\mid+\mid\frac{\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{3}} }{\left(\mathrm{n}+\mathrm{2}\right)!}\mid+...+\mid\frac{\left(−\mathrm{1}\right)^{\mathrm{m}+\mathrm{1}} }{\mathrm{m}!}\mid \\ $$$$=\frac{\mathrm{1}}{\left(\mathrm{n}+\mathrm{1}\right)!}+\frac{\mathrm{1}}{\left(\mathrm{n}+\mathrm{2}\right)!}+...+\frac{\mathrm{1}}{\mathrm{m}!}<\frac{\mathrm{1}}{\mathrm{n}!}\leqslant\frac{\mathrm{1}}{\mathrm{n}}<\epsilon. \\ $$$$\frac{\mathrm{1}}{\mathrm{n}}\rightarrow\mathrm{0}\:\mathrm{as}\:\mathrm{n}\rightarrow\infty,\:\mathrm{so}\:\mathrm{no}\:\mathrm{matter}\:\mathrm{any}\:\epsilon>\mathrm{0}\:\frac{\mathrm{1}}{\mathrm{n}}<\epsilon\:\mathrm{eventually}. \\ $$$$\mathrm{as}\:\mathrm{long}\:\mathrm{as}\:\mathrm{n}^{\ast} >\frac{\mathrm{1}}{\epsilon}\:\mid\mathrm{a}_{\mathrm{m}} −\mathrm{a}_{\mathrm{n}} \mid<\epsilon\:\forall\mathrm{n}\geqslant\mathrm{n}^{\ast} .\: \\ $$

Question Number 148638    Answers: 0   Comments: 1

find laurent series f(z)=((cos(iz))/z^n ) ,∣z−i∣>2

$${find}\:{laurent}\:{series}\:{f}\left({z}\right)=\frac{{cos}\left({iz}\right)}{{z}^{{n}} }\:\:,\mid{z}−{i}\mid>\mathrm{2} \\ $$

Question Number 148633    Answers: 0   Comments: 0

Question Number 148630    Answers: 1   Comments: 0

Question Number 148631    Answers: 1   Comments: 1

find the region converge of the series and find the sum Σ_(n=1) ^∞ (n/(2^n (z−1)^n )) ?

$${find}\:{the}\:{region}\:{converge}\:{of}\:{the}\:{series}\:{and}\: \\ $$$${find}\:{the}\:{sum}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}}{\mathrm{2}^{{n}} \left({z}−\mathrm{1}\right)^{{n}} }\:? \\ $$

Question Number 148620    Answers: 1   Comments: 0

consider the following pdf of a random variable X f(x)={Σ_(i=0) ^∞ [(−x^2 )i/i!]_(0 otherwise) ^(x>0) find the variance X

$$\mathrm{consider}\:\mathrm{the}\:\mathrm{following}\:\mathrm{pdf}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{random}\:\mathrm{variable}\:\mathrm{X} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\left\{\sum_{\mathrm{i}=\mathrm{0}} ^{\infty} \left[\left(−\mathrm{x}^{\mathrm{2}} \right)\mathrm{i}/\mathrm{i}!\right]_{\mathrm{0}\:\mathrm{otherwise}} ^{\mathrm{x}>\mathrm{0}} \:\right. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{variance}\:\mathrm{X} \\ $$$$ \\ $$

Question Number 148609    Answers: 0   Comments: 2

prove that (a_n )_(n≥1 ) defined by a_n =(1/2)+(1/6)+...+(1/(n(n+1))) is cauchy sequence.

$$\mathrm{prove}\:\mathrm{that}\:\left(\mathrm{a}_{\mathrm{n}} \right)_{\mathrm{n}\geqslant\mathrm{1}\:} \mathrm{defined}\:\mathrm{by}\:\mathrm{a}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{6}}+...+\frac{\mathrm{1}}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)}\:\mathrm{is}\: \\ $$$$\mathrm{cauchy}\:\mathrm{sequence}. \\ $$

Question Number 148600    Answers: 1   Comments: 0

lim_(x→−∞) ((27x^3 −3x^2 ))^(1/3) +((8x^3 −x^2 ))^(1/3) −((x^3 −4x^2 +2021))^(1/3) =?

$$\:\:\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\sqrt[{\mathrm{3}}]{\mathrm{27}{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} }\:+\sqrt[{\mathrm{3}}]{\mathrm{8}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} }−\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} −\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2021}}\:=?\: \\ $$

Question Number 148599    Answers: 3   Comments: 0

f : Z → Z f(x) = 2 ∙ f(x - 1) f(5) = 4 find f(30) = ?

$${f}\::\:\mathbb{Z}\:\rightarrow\:\mathbb{Z} \\ $$$${f}\left({x}\right)\:=\:\mathrm{2}\:\centerdot\:{f}\left({x}\:-\:\mathrm{1}\right) \\ $$$${f}\left(\mathrm{5}\right)\:=\:\mathrm{4} \\ $$$${find}\:\:\:{f}\left(\mathrm{30}\right)\:=\:? \\ $$

Question Number 148594    Answers: 2   Comments: 2

Solve for equation: x^2 +y^2 +z^2 = xy+xz+yz ⇒ x;y;z=?

$${Solve}\:{for}\:{equation}: \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:=\:{xy}+{xz}+{yz}\:\:\:\Rightarrow\:\:{x};{y};{z}=? \\ $$

Question Number 148573    Answers: 2   Comments: 0

log_(√x) (x/y) = A ⇒ log_(√y) (y/x) = ?

$${log}_{\sqrt{\boldsymbol{{x}}}} \:\frac{{x}}{{y}}\:=\:{A}\:\:\Rightarrow\:\:{log}_{\sqrt{\boldsymbol{{y}}}} \:\frac{{y}}{{x}}\:=\:? \\ $$

Question Number 148654    Answers: 4   Comments: 0

lim_(x→1) (((1+x)/(2+x)))^((1−(√x))/(1−x)) = ?

$$\underset{{x}\rightarrow\mathrm{1}} {{lim}}\left(\frac{\mathrm{1}+{x}}{\mathrm{2}+{x}}\right)^{\frac{\mathrm{1}−\sqrt{\boldsymbol{{x}}}}{\mathrm{1}−\boldsymbol{{x}}}} \:=\:? \\ $$

Question Number 148653    Answers: 0   Comments: 0

Question Number 148570    Answers: 2   Comments: 0

calculate ∫_0 ^∞ ((logx)/(x^2 +x+1))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{logx}}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 148569    Answers: 2   Comments: 1

Question Number 148568    Answers: 2   Comments: 0

calculate lim_(x→0) ((sh(2sinx)−sin(sh(2x)))/x^2 )

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\:\frac{\mathrm{sh}\left(\mathrm{2sinx}\right)−\mathrm{sin}\left(\mathrm{sh}\left(\mathrm{2x}\right)\right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 148567    Answers: 1   Comments: 0

find ∫_0 ^∞ ((x^2 logx)/((x^2 +1)^3 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} \:\mathrm{logx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }\mathrm{dx} \\ $$

Question Number 148566    Answers: 0   Comments: 0

calculate U_n =∫∫_([(1/n),n[) ((cos(x^2 +y^2 ))/(x^2 +y^2 ))dxdy and determine lim_(n→+∞) U_n nature of Σ U_n ?

$$\mathrm{calculate}\:\:\mathrm{U}_{\mathrm{n}} =\int\int_{\left[\frac{\mathrm{1}}{\mathrm{n}},\mathrm{n}\left[\right.\right.} \:\:\:\frac{\mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} }\mathrm{dxdy} \\ $$$$\mathrm{and}\:\mathrm{determine}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{U}_{\mathrm{n}} \\ $$$$\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} ? \\ $$

Question Number 148565    Answers: 2   Comments: 0

calculate ∫_(−∞) ^(+∞) ((x^2 dx)/((x^2 −x+1)(x^2 +x+1)))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}\right)} \\ $$

Question Number 148564    Answers: 1   Comments: 0

calculate ∫_1 ^2 ((logx)/(1+x))dx

$$\mathrm{calculate}\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\frac{\mathrm{logx}}{\mathrm{1}+\mathrm{x}}\mathrm{dx} \\ $$

Question Number 148559    Answers: 1   Comments: 0

Question Number 148558    Answers: 2   Comments: 0

Trouver toutes les fonctions continues f:R→R verifiant: ∀(x,y)∈R^2 , f(x+y)f(x−y)=f^2 (x)f^2 (y).. monsieur j′ai suppose^ que f est un morphisme mutiplicatif de R.. mais ca ne sort pas...

$$\mathrm{Trouver}\:\mathrm{toutes}\:\mathrm{les}\:\mathrm{fonctions}\:\mathrm{continues} \\ $$$$\mathrm{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{verifiant}: \\ $$$$\forall\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{R}^{\mathrm{2}} ,\:\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)\mathrm{f}\left(\mathrm{x}−\mathrm{y}\right)=\mathrm{f}^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{f}^{\mathrm{2}} \left(\mathrm{y}\right).. \\ $$$$\mathrm{monsieur}\:\mathrm{j}'\mathrm{ai}\:\mathrm{suppos}\acute {\mathrm{e}}\:\mathrm{que}\:\mathrm{f}\:\mathrm{est}\:\mathrm{un}\: \\ $$$$\mathrm{morphisme}\:\mathrm{mutiplicatif}\:\mathrm{de}\:\mathbb{R}..\:\mathrm{mais}\:\mathrm{ca}\:\mathrm{ne} \\ $$$$\mathrm{sort}\:\mathrm{pas}... \\ $$

Question Number 148550    Answers: 1   Comments: 2

A(0;0) ; B(−2;4) and C(−6;14) if the triangle has vertices, find the lenght of the median drawn from the vertx C

$$\boldsymbol{{A}}\left(\mathrm{0};\mathrm{0}\right)\:\:;\:\:\boldsymbol{{B}}\left(−\mathrm{2};\mathrm{4}\right)\:\:{and}\:\:\boldsymbol{{C}}\left(−\mathrm{6};\mathrm{14}\right) \\ $$$${if}\:{the}\:{triangle}\:{has}\:{vertices},\:{find}\:{the} \\ $$$${lenght}\:{of}\:{the}\:{median}\:{drawn}\:{from} \\ $$$${the}\:{vertx}\:\boldsymbol{{C}} \\ $$

Question Number 148552    Answers: 0   Comments: 2

Question Number 148546    Answers: 3   Comments: 2

lim_(x→0) (((√(x + 4)) - 2)/(sinx)) = ?

$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {{lim}}\:\frac{\sqrt{{x}\:+\:\mathrm{4}}\:-\:\mathrm{2}}{{sinx}}\:=\:? \\ $$

Question Number 148543    Answers: 4   Comments: 0

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