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Question Number 120534 Answers: 2 Comments: 1

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Let a>0 , g(x)=Σ_(n∈Z) exp(−(((x−n)^2 )/a)) Find the Fourier coefficients of g and deduce that for x∈R^∗ Σ_(n∈Z) exp(−((πn^2 )/x^2 ))= x .Σ_(n∈Z) exp(−πn^2 x^2 )

$${Let}\:\:{a}>\mathrm{0}\:,\:{g}\left({x}\right)=\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\frac{\left({x}−{n}\right)^{\mathrm{2}} }{{a}}\right) \\ $$$${Find}\:{the}\:{Fourier}\:{coefficients}\:{of}\:\:\:{g} \\ $$$${and}\:{deduce}\:{that}\:{for}\:{x}\in\mathbb{R}^{\ast} \\ $$$$\:\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\frac{\pi{n}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)=\:{x}\:.\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\pi{n}^{\mathrm{2}} {x}^{\mathrm{2}} \right) \\ $$$$ \\ $$

Question Number 120524 Answers: 0 Comments: 0

1/.Given E=C([0,1],R) are mapping set [0,1]→R. (a). Show that E is vector space. (b).For f∈E ,let ∣∣f∣∣_1 =∫_0 ^1 ∣f(x)∣dx . Show that (E,∣∣.∣∣_1 ) are normed space. (Helpe me please)

$$\mathrm{1}/.\mathrm{Given}\:\mathrm{E}=\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\mathbb{R}\right)\:\mathrm{are}\:\mathrm{mapping}\:\mathrm{set}\:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}. \\ $$$$\left(\mathrm{a}\right).\:\mathrm{Show}\:\mathrm{that}\:\mathrm{E}\:\mathrm{is}\:\mathrm{vector}\:\mathrm{space}. \\ $$$$\left(\mathrm{b}\right).\mathrm{For}\:\:\mathrm{f}\in\mathrm{E}\:,\mathrm{let}\:\mid\mid\mathrm{f}\mid\mid_{\mathrm{1}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mid\mathrm{f}\left(\mathrm{x}\right)\mid\mathrm{dx}\:. \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{E},\mid\mid.\mid\mid_{\mathrm{1}} \right)\:\mathrm{are}\:\mathrm{normed}\:\mathrm{space}. \\ $$$$\:\:\left(\mathrm{Helpe}\:\mathrm{me}\:\mathrm{please}\right) \\ $$

Question Number 120519 Answers: 1 Comments: 1

Question Number 120515 Answers: 1 Comments: 0

lim_(x→∞) (3 (2)^(1/(x )) −2.(3)^(1/(x )) )^x ?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{3}\:\sqrt[{{x}\:}]{\mathrm{2}}\:−\mathrm{2}.\sqrt[{{x}\:}]{\mathrm{3}}\:\right)^{{x}} \:? \\ $$

Question Number 120511 Answers: 3 Comments: 0

lim_(x→0) ((√([ sin^(−1) (2x)]^3 ))/(x sin ((√x)))) ?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\left[\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{2}{x}\right)\right]^{\mathrm{3}} }}{{x}\:\mathrm{sin}\:\left(\sqrt{{x}}\right)}\:?\: \\ $$

Question Number 120502 Answers: 2 Comments: 0

It takes Musa 3 days to buid a room while it takes John 4 days. How many days will it takes both of them?

$$\mathrm{It}\:\mathrm{takes}\:\mathrm{Musa}\:\mathrm{3}\:\mathrm{days}\:\mathrm{to}\:\mathrm{buid}\:\mathrm{a}\:\mathrm{room} \\ $$$$\mathrm{while}\:\mathrm{it}\:\mathrm{takes}\:\mathrm{John}\:\mathrm{4}\:\mathrm{days}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{days}\:\mathrm{will}\:\mathrm{it}\:\mathrm{takes}\:\mathrm{both}\:\mathrm{of}\:\mathrm{them}? \\ $$

Question Number 120480 Answers: 4 Comments: 0

show that ∀ n ∈N^∗ Σ_(k=1) ^n k(n−k)=(((n−1)(n+1))/6)

$${show}\:{that}\:\forall\:{n}\:\in\mathbb{N}^{\ast} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left({n}−{k}\right)=\frac{\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$

Question Number 120477 Answers: 2 Comments: 0

Question Number 120475 Answers: 1 Comments: 0

Question Number 120472 Answers: 0 Comments: 0

A=5x23^(−) ^6 show that A≡x−4[7] deduct the value of x for which A is divisible by 7

$${A}=\overline {\mathrm{5}{x}\mathrm{23}}\:^{\mathrm{6}} \:{show}\:{that}\:{A}\equiv{x}−\mathrm{4}\left[\mathrm{7}\right] \\ $$$${deduct}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{A}\:{is}\: \\ $$$${divisible}\:{by}\:\mathrm{7} \\ $$

Question Number 120471 Answers: 1 Comments: 0

solve in Z x^3 +2x+1≡1[4]

$${solve}\:{in}\:\mathbb{Z}\:{x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{1}\equiv\mathrm{1}\left[\mathrm{4}\right] \\ $$

Question Number 120470 Answers: 0 Comments: 0

solve in function of n: 2^n ≡x−4[3] n∈N

$${solve}\:{in}\:{function}\:{of}\:{n}: \\ $$$$\mathrm{2}^{{n}} \equiv{x}−\mathrm{4}\left[\mathrm{3}\right] \\ $$$${n}\in\mathbb{N} \\ $$

Question Number 120469 Answers: 1 Comments: 0

c alculate the rest of the division of 2^n by 3 ; n ∈ N

$${c}\:{alculate}\:{the}\:{rest}\:{of}\:{the}\:{division}\:{of} \\ $$$$\mathrm{2}^{{n}} \:{by}\:\mathrm{3}\:;\:{n}\:\in\:\mathbb{N} \\ $$

Question Number 120468 Answers: 0 Comments: 2

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Question Number 120464 Answers: 1 Comments: 0

Question Number 120463 Answers: 2 Comments: 0

Find the slope of the line tangent to the graph r=3cos^2 (2θ) at θ=(π/6).

$${Find}\:{the}\:{slope}\:{of}\:{the}\:{line}\:{tangent} \\ $$$${to}\:{the}\:{graph}\:{r}=\mathrm{3cos}\:^{\mathrm{2}} \left(\mathrm{2}\theta\right)\:{at} \\ $$$$\theta=\frac{\pi}{\mathrm{6}}.\: \\ $$

Question Number 120461 Answers: 1 Comments: 0

∫sin(lnx)dx

$$\int\mathrm{sin}\left(\mathrm{ln}{x}\right){dx} \\ $$

Question Number 120459 Answers: 0 Comments: 1

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Question Number 120452 Answers: 2 Comments: 0

Find the equation of the line tangent to the parametric curve given by the equations { ((x=(1+t^3 )^4 +t^2 )),((y=t^5 +t^2 +2)) :} at t=1

$${Find}\:{the}\:{equation}\:{of}\:{the}\:{line}\:{tangent}\:{to}\:{the}\:{parametric} \\ $$$${curve}\:{given}\:{by}\:{the}\:{equations}\:\begin{cases}{{x}=\left(\mathrm{1}+{t}^{\mathrm{3}} \right)^{\mathrm{4}} +{t}^{\mathrm{2}} }\\{{y}={t}^{\mathrm{5}} +{t}^{\mathrm{2}} +\mathrm{2}}\end{cases}\:{at}\:{t}=\mathrm{1} \\ $$

Question Number 120445 Answers: 1 Comments: 0

Question Number 120444 Answers: 0 Comments: 0

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