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Question Number 41410 Answers: 0 Comments: 2

let A_n =∫_0 ^∞ [ne^(−x) ]dx with n≥2 1) calculate A_n 2) find nature of Σ_(n≥2) A_n 3) study the convergence of Σ (1/A_n ) and Σ (1/A_n ^2 )

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\left[{ne}^{−{x}} \right]{dx}\:\:{with}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\sum_{{n}\geqslant\mathrm{2}} \:\:\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:\:\Sigma\:\frac{\mathrm{1}}{{A}_{{n}} }\:\:{and}\:\Sigma\:\frac{\mathrm{1}}{{A}_{{n}} ^{\mathrm{2}} } \\ $$

Question Number 41409 Answers: 1 Comments: 0

calculate S_p =Σ_(n=1) ^∞ (((−1)^n )/(n(n+1)(n+2)...(n+p))) with p fromN

$${calculate}\:{S}_{{p}} =\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)...\left({n}+{p}\right)}\:\:{with}\:{p}\:{fromN} \\ $$

Question Number 41408 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (((−1)^n )/(n(n+1)(n+2)(n+3)))

$${calculate}\:\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)} \\ $$

Question Number 41407 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (((−1)^n )/(n(n+1)(n+2)))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$

Question Number 41394 Answers: 1 Comments: 1

show that a) ((1−cos2A)/(1+cos2A)) ≡ tan^2 A b) ((sin2A)/(1+cos2A))≡ tanA.

$$\left.{show}\:{that}\:{a}\right)\:\frac{\mathrm{1}−{cos}\mathrm{2}{A}}{\mathrm{1}+{cos}\mathrm{2}{A}}\:\equiv\:{tan}^{\mathrm{2}} {A} \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}\right)\:\frac{{sin}\mathrm{2}{A}}{\mathrm{1}+{cos}\mathrm{2}{A}}\equiv\:{tanA}. \\ $$

Question Number 41392 Answers: 0 Comments: 1

If both A− (I/2) and A+(I/2) are orthogonal matrices, then prove that A^2 = −(3/4) I.

$$\mathrm{If}\:\mathrm{both}\:\mathrm{A}−\:\frac{\mathrm{I}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{A}+\frac{\mathrm{I}}{\mathrm{2}}\:\mathrm{are}\:\mathrm{orthogonal} \\ $$$$\mathrm{matrices},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\: \\ $$$$\mathrm{A}^{\mathrm{2}} =\:−\frac{\mathrm{3}}{\mathrm{4}}\:\mathrm{I}. \\ $$

Question Number 41390 Answers: 1 Comments: 0

solve 2^(x−2) =8x

$$\boldsymbol{\mathrm{solve}}\:\:\mathrm{2}^{\boldsymbol{{x}}−\mathrm{2}} =\mathrm{8}\boldsymbol{{x}} \\ $$

Question Number 41389 Answers: 1 Comments: 2

if tan^2 A+tan^2 B+3=0 show that cos^2 B+2cos^2 A=0

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{tan}}^{\mathrm{2}} \boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{tan}}^{\mathrm{2}} \boldsymbol{\mathrm{B}}+\mathrm{3}=\mathrm{0} \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{B}}+\mathrm{2}\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{A}}=\mathrm{0} \\ $$

Question Number 41388 Answers: 0 Comments: 1

Question Number 41381 Answers: 2 Comments: 4

Question Number 41378 Answers: 1 Comments: 1

Solve : e^x (x+1)dx + (ye^y − xe^x )dy=0

$$\mathrm{Solve}\:: \\ $$$$\mathrm{e}^{{x}} \left({x}+\mathrm{1}\right){dx}\:+\:\left(\mathrm{ye}^{\mathrm{y}} \:−\:{xe}^{{x}} \right)\mathrm{dy}=\mathrm{0} \\ $$

Question Number 41371 Answers: 1 Comments: 1

Calculate the amount of energy⊛ released when carbon-12 nucleus is formed by bringing 6 neutrons and 6 protons together.

$${Calculate}\:{the}\:{amount}\:{of}\:{energy}\circledast \\ $$$${released}\:{when}\:{carbon}-\mathrm{12}\:{nucleus} \\ $$$${is}\:{formed}\:{by}\:{bringing}\:\mathrm{6}\:{neutrons} \\ $$$${and}\:\mathrm{6}\:{protons}\:{together}. \\ $$

Question Number 41369 Answers: 0 Comments: 0

Question Number 41373 Answers: 0 Comments: 1

when uranium-235 undergoes nuclear fission,0.01% of its mass is converted to energy.(a)Calculate the energy released when 1gm of Uranium-235 undergoes nuclear fission.(b)What is the amount of uranium-235 that will undergo nuclear fission reaction every hour in a nuclear reactor that provides 100 megawatts of electric power?

$${when}\:{uranium}-\mathrm{235}\:{undergoes}\: \\ $$$${nuclear}\:{fission},\mathrm{0}.\mathrm{01\%}\:{of}\:{its}\:{mass} \\ $$$${is}\:{converted}\:{to}\:{energy}.\left({a}\right){Calculate} \\ $$$${the}\:{energy}\:{released}\:{when}\:\mathrm{1}{gm}\:{of} \\ $$$${Uranium}-\mathrm{235}\:{undergoes}\:{nuclear} \\ $$$${fission}.\left({b}\right){What}\:{is}\:{the}\:{amount}\:{of} \\ $$$${uranium}-\mathrm{235}\:{that}\:{will}\:{undergo} \\ $$$${nuclear}\:{fission}\:{reaction}\:{every}\:{hour} \\ $$$${in}\:{a}\:{nuclear}\:{reactor}\:{that}\:{provides} \\ $$$$\mathrm{100}\:{megawatts}\:{of}\:{electric}\:{power}? \\ $$

Question Number 41349 Answers: 0 Comments: 2

calculate Σ_(n=1) ^∞ (((−1)^n )/(3n−1))

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{3}{n}−\mathrm{1}} \\ $$

Question Number 41348 Answers: 0 Comments: 5

Question Number 41347 Answers: 1 Comments: 0

Question Number 41346 Answers: 0 Comments: 6

calculate ∫∫_([0,1]^2 ) cos(x^2 +y^2 )dxdy .

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:{cos}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\:. \\ $$

Question Number 41345 Answers: 0 Comments: 2

let u_n = Σ_(k=1) ^n (1/k) −ln(n) 1) prove that (u_n )is convergent 2) let γ =lim_(n→+∞) u_n prove that 0<γ<1

$${let}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:−{ln}\left({n}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{convergent} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\gamma\:={lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:\:\:{prove}\:{that}\:\mathrm{0}<\gamma<\mathrm{1}\:\: \\ $$

Question Number 41343 Answers: 1 Comments: 1

calculate ∫∫_D (x^2 −y^2 )dxdy with D = [−1,1]^2

$${calculate}\:\:\:\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right){dxdy}\:\:{with} \\ $$$${D}\:=\:\left[−\mathrm{1},\mathrm{1}\right]^{\mathrm{2}} \\ $$

Question Number 41342 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)^2 (n+2)^2 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$

Question Number 41361 Answers: 1 Comments: 0

The half life of radium-226 is 1620 years.Calculate (a)the decay constant (b)the time it takes 60% of a given sample to decay, and (c)the initial activity of 1gm of pure radium-226

$${The}\:{half}\:{life}\:{of}\:{radium}-\mathrm{226}\:{is}\:\mathrm{1620} \\ $$$${years}.{Calculate}\:\left({a}\right){the}\:{decay}\:{constant} \\ $$$$\left({b}\right){the}\:{time}\:{it}\:{takes}\:\mathrm{60\%}\:{of}\:{a}\:{given} \\ $$$${sample}\:{to}\:{decay},\:{and}\:\left({c}\right){the}\:{initial} \\ $$$${activity}\:{of}\:\mathrm{1}{gm}\:{of}\:{pure}\:{radium}-\mathrm{226} \\ $$

Question Number 41332 Answers: 1 Comments: 3

In Matrices, Is (A^(−1) )B = B(A^(−1) ) ?

$$\mathrm{In}\:\mathrm{Matrices},\: \\ $$$$\mathrm{Is}\:\left(\mathrm{A}^{−\mathrm{1}} \right)\mathrm{B}\:=\:\mathrm{B}\left(\mathrm{A}^{−\mathrm{1}} \right)\:? \\ $$

Question Number 41327 Answers: 0 Comments: 0

Derive the sum of an Harmonic Progression

$${Derive}\:{the}\:{sum}\:{of}\:{an}\:{Harmonic}\:{Progression} \\ $$

Question Number 41326 Answers: 0 Comments: 0

Evaluate ∫_0 ^(π/2) x^3 sec^5 x dx

$${Evaluate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{3}} {sec}^{\mathrm{5}} {x}\:{dx} \\ $$

Question Number 41324 Answers: 1 Comments: 2

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