Question and Answers Forum

let give I_n = ∫_0 ^∞ (dt/((1+t^2 )^n )) with n integr and n≥1 1) prove the convergence of I_n 2)find lim_(n→∞) I_n 3) study the convergence of the serie Σ_(n=1) ^∞ (−1)^n I_n .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{{n}} }\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{convergence}\:{of}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow\infty} \:\:{I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:{the}\:{serie}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:\:{I}_{{n}} \:. \\ $$

Question Number 31965 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ (((−1)^(n−1) −2^n )/n) x^n with ∣x∣ <(1/2)

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:−\mathrm{2}^{{n}} }{{n}}\:{x}^{{n}} \:\:{with}\:\mid{x}\mid\:<\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 31964 Answers: 0 Comments: 0

1)find S_n = Σ_(k=0) ^n C_n ^k sin((k/n)) 2) study the convergence of S_n

$$\left.\mathrm{1}\right){find}\:\:{S}_{{n}} \:\:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:{sin}\left(\frac{{k}}{{n}}\right) \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:{S}_{{n}} \\ $$

Question Number 31963 Answers: 0 Comments: 0

find Re (((1+e^(iα) )/(1+e^(iβ) ))) and Im ( ((1+e^(iα) )/(1+e^(iβ) )) ) .

$${find}\:{Re}\:\left(\frac{\mathrm{1}+{e}^{{i}\alpha} }{\mathrm{1}+{e}^{{i}\beta} }\right)\:{and}\:{Im}\:\left(\:\frac{\mathrm{1}+{e}^{{i}\alpha} }{\mathrm{1}+{e}^{{i}\beta} }\:\right)\:. \\ $$

Question Number 31962 Answers: 0 Comments: 0

let f(x)= (e^(2x) /(x+1)) 1) calculate f^((n)) (x) 2) find f^((n)) (o) and f^((n)) (1) .

$${let}\:{f}\left({x}\right)=\:\frac{{e}^{\mathrm{2}{x}} }{{x}+\mathrm{1}}\:\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left({o}\right)\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{1}\right)\:. \\ $$

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If a, b, c, d are in GP, then (a^3 +b^3 )^(−1) , (b^3 +c^3 )^(−1) , (c^3 +a^3 )^(−1) are in

$$\mathrm{If}\:{a},\:{b},\:{c},\:{d}\:\mathrm{are}\:\mathrm{in}\:\mathrm{GP},\:\mathrm{then}\:\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} \right)^{−\mathrm{1}} ,\: \\ $$$$\left({b}^{\mathrm{3}} +{c}^{\mathrm{3}} \right)^{−\mathrm{1}} ,\:\left({c}^{\mathrm{3}} +{a}^{\mathrm{3}} \right)^{−\mathrm{1}} \:\mathrm{are}\:\mathrm{in} \\ $$

Question Number 31952 Answers: 0 Comments: 1

For a sequence < a_n > , a_1 = 2 and (a_(n+1) /a_n ) = (1/3) . Then Σ_(r=1) ^(20) a_r is

$$\mathrm{For}\:\mathrm{a}\:\mathrm{sequence}\:<\:{a}_{{n}} \:>\:\:,\:{a}_{\mathrm{1}} =\:\mathrm{2}\:\mathrm{and}\: \\ $$$$\frac{{a}_{{n}+\mathrm{1}} }{{a}_{{n}} }\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\:.\:\:\mathrm{Then}\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\:{a}_{{r}} \:\mathrm{is} \\ $$

Question Number 31951 Answers: 1 Comments: 0

Evaluate ∫sin (√x)dx

$${Evaluate}\:\int\mathrm{sin}\:\sqrt{{x}}{dx} \\ $$

Question Number 31949 Answers: 1 Comments: 0

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Calculate Σ_(j≤k≤i) (−1)^k ((i),(k) ) ((k),(j) )

$$\mathrm{Calculate}\:\underset{{j}\leqslant{k}\leqslant{i}} {\Sigma}\:\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{i}}\\{{k}}\end{pmatrix}\begin{pmatrix}{{k}}\\{{j}}\end{pmatrix} \\ $$

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The rigid body of mass 10kg rotates such that the radius of the circular path it describes is 5cm.It it moves from the point P to Q in 5s covering an angular distance of 30°,Calculate : a)the final angular velocity if the angular acceleration is 4m/s^2 b)The moment of inertia of the body. c)angular momentum d)rotational kinetic energy

$${The}\:{rigid}\:{body}\:{of}\:{mass}\:\mathrm{10}{kg} \\ $$$${rotates}\:{such}\:{that}\:{the}\:{radius}\:{of}\:{the} \\ $$$${circular}\:{path}\:{it}\:{describes}\:{is}\:\mathrm{5}{cm}.{It} \\ $$$${it}\:{moves}\:{from}\:{the}\:{point}\:{P}\:{to}\:{Q}\:{in} \\ $$$$\mathrm{5}{s}\:{covering}\:{an}\:{angular}\:{distance} \\ $$$${of}\:\mathrm{30}°,{Calculate}\:: \\ $$$$\left.{a}\right){the}\:{final}\:{angular}\:{velocity}\:{if} \\ $$$${the}\:{angular}\:{acceleration}\:{is}\:\mathrm{4}{m}/{s}^{\mathrm{2}} \\ $$$$\left.{b}\right){The}\:{moment}\:{of}\:{inertia}\:{of}\:{the} \\ $$$${body}. \\ $$$$\left.{c}\right){angular}\:{momentum} \\ $$$$\left.{d}\right){rotational}\:{kinetic}\:{energy} \\ $$

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