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Question Number 30764 Answers: 0 Comments: 1

let I_n = ∫_0 ^(1/2) (1−2t)^n e^(−t) dt with n integr not 0 1) prove that ∀t∈[0,(1/2)] (1/(√e))(1−2t)^n ≤ (1−2t)^n e^(−t) ≤(1−2t)^n then find lim_(n→∞ ) I_n 2) prove that I_(n+1 ) =1−2(n+1)I_n 3) calculate I_1 ,I_2 , and I_3 .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} {dt}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall{t}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$\frac{\mathrm{1}}{\sqrt{{e}}}\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \leqslant\:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} \:\leqslant\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{then}\:{find}\:{lim}_{{n}\rightarrow\infty\:} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{I}_{{n}+\mathrm{1}\:} =\mathrm{1}−\mathrm{2}\left({n}+\mathrm{1}\right){I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,\:{and}\:{I}_{\mathrm{3}} . \\ $$

Question Number 30761 Answers: 0 Comments: 1

find ∫_0 ^∞ x^(2n+1) e^(−x^2 ) dx with n from N.

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:\:\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 30760 Answers: 0 Comments: 1

find I_n = ∫_0 ^1 (lnx)^n dx with n fromN

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left({lnx}\right)^{{n}} \:{dx}\:\:{with}\:{n}\:{fromN} \\ $$

Question Number 30759 Answers: 0 Comments: 1

find lim_(n→∞) (((n!)/n^n ))^(1/n) .

$${find}\:{lim}_{{n}\rightarrow\infty} \:\:\:\:\left(\frac{{n}!}{{n}^{{n}} }\right)^{\frac{\mathrm{1}}{{n}}} \:. \\ $$

Question Number 30758 Answers: 0 Comments: 1

find lim_(n→∞) ((1/(n+1)) +(1/(n+2)) +....+(1/(n+p))) pfixed fromN^★

$${find}\:{lim}_{{n}\rightarrow\infty} \left(\frac{\mathrm{1}}{{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{{n}+\mathrm{2}}\:+....+\frac{\mathrm{1}}{{n}+{p}}\right)\:{pfixed}\:{fromN}^{\bigstar} \\ $$$$ \\ $$

Question Number 30757 Answers: 0 Comments: 1

find lim_(n→∞) ((1/n) +(1/(√(n^2 −1))) +.... +(1/(√(n^2 −(n−1)^2 ))) )

$${find}\:{lim}_{{n}\rightarrow\infty} \:\:\left(\frac{\mathrm{1}}{{n}}\:+\frac{\mathrm{1}}{\sqrt{{n}^{\mathrm{2}} \:−\mathrm{1}}}\:+....\:+\frac{\mathrm{1}}{\sqrt{{n}^{\mathrm{2}} \:−\left({n}−\mathrm{1}\right)^{\mathrm{2}} }}\:\right) \\ $$

Question Number 30755 Answers: 0 Comments: 1

find lim_(n→∞) ^n (√((1+(1/n))(1+(2/n))...(1+(n/n)) ))

$${find}\:\:\:{lim}_{{n}\rightarrow\infty} \:\:^{{n}} \sqrt{\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\left(\mathrm{1}+\frac{\mathrm{2}}{{n}}\right)...\left(\mathrm{1}+\frac{{n}}{{n}}\right)\:} \\ $$

Question Number 30754 Answers: 0 Comments: 0

prove that e^x =Σ_(k=0) ^n (x^k /(k!)) +(x^(n+1) /(n!)) ∫_0 ^ (1−t)^n e^(tx) dt 2) prove that e^x = Σ_(k=0) ^(∞ ) (x^k /(k!)) .

$${prove}\:{that}\:{e}^{{x}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{{x}^{{k}} }{{k}!}\:+\frac{{x}^{{n}+\mathrm{1}} }{{n}!}\:\int_{\mathrm{0}} ^{} \left(\mathrm{1}−{t}\right)^{{n}} \:{e}^{{tx}} {dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:{e}^{{x}} =\:\sum_{{k}=\mathrm{0}} ^{\infty\:} \:\:\frac{{x}^{{k}} }{{k}!}\:. \\ $$

Question Number 30753 Answers: 0 Comments: 0

let f_n (x)=(1/x^(n+1) ) (e^x −Σ_(p=0) ^(n ) (x^p /(p!))) 1) prove that f^((n)) (x)=((Q_n (x) e^x −P_n (x))/x^(2n+1) ) find the polynomial P_n and Q_n . 2) prove that e^x −((P_n (x))/(Q_n (x)))=o(x^(2n+1) )

$${let}\:{f}_{{n}} \left({x}\right)=\frac{\mathrm{1}}{{x}^{{n}+\mathrm{1}} }\:\left({e}^{{x}} \:\:−\sum_{{p}=\mathrm{0}} ^{{n}\:} \:\:\:\frac{{x}^{{p}} }{{p}!}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)=\frac{{Q}_{{n}} \left({x}\right)\:{e}^{{x}} \:−{P}_{{n}} \left({x}\right)}{{x}^{\mathrm{2}{n}+\mathrm{1}} }\:{find}\:{the} \\ $$$${polynomial}\:{P}_{{n}} \:{and}\:{Q}_{{n}} . \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{e}^{{x}} \:−\frac{{P}_{{n}} \left({x}\right)}{{Q}_{{n}} \left({x}\right)}={o}\left({x}^{\mathrm{2}{n}+\mathrm{1}} \:\:\right) \\ $$

Question Number 30752 Answers: 0 Comments: 1

let f(x)= (1/(1+x^2 )) calculate f^((n)) (x).

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

Question Number 30751 Answers: 0 Comments: 0

study and give the graph for f(x)=(x^2 /(x−1)) e^(1/x) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for}\: \\ $$$${f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}−\mathrm{1}}\:{e}^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 30750 Answers: 0 Comments: 0

f function C^∞ /f^′ =1+f^2 let take a_k =((f^((k)) (0))/(k!)) prove that a_(n+1) = (1/(n+1)) Σ_(k=0) ^n a_k a_(n−k)

$${f}\:{function}\:{C}^{\infty} \:/{f}^{'} =\mathrm{1}+{f}^{\mathrm{2}} \:\:{let}\:{take}\:{a}_{{k}} =\frac{{f}^{\left({k}\right)} \left(\mathrm{0}\right)}{{k}!}\: \\ $$$${prove}\:{that}\:{a}_{{n}+\mathrm{1}} =\:\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{a}_{{k}} \:\:{a}_{{n}−{k}} \\ $$

Question Number 30749 Answers: 0 Comments: 0

let f(x)=arcsinx with x∈[0,1] 1) prove that (1−x^2 )f^(′′) (x) −xf^′ (x)=0 2)prove that (1−x^2 )f^((n+2)) (x)=(2n+1)x f^((n+1)) (x) +n^2 f^((n)) (x) 3) prove that f^((n)) (x) ≥0 ∀n .

$${let}\:{f}\left({x}\right)={arcsinx}\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{''} \left({x}\right)\:−{xf}^{'} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)=\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+{n}^{\mathrm{2}} {f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left({n}\right)} \left({x}\right)\:\geqslant\mathrm{0}\:\forall{n}\:. \\ $$

Question Number 30748 Answers: 0 Comments: 0

let a>0 and b>0 find lim_(x→0^+ ) ( ((a^x +b^x )/2))^(1/x) .

$${let}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\left(\:\:\frac{{a}^{{x}} \:+{b}^{{x}} }{\mathrm{2}}\right)^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 30747 Answers: 0 Comments: 0

let f(x)=(√(1+x^2 )) 1)find a d.e.wich verify f(x) 2) prove that ∀x∈R ,∀n∈N (1+x^2 )f^((n+2)) (x)+(2n+1)x f^((n+1)) (x) +(n^2 −1)f^((n)) (x)=0 3) prove that f^((2n+1)) (0)=0 ∀n∈ N

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{d}.{e}.{wich}\:{verify}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall{x}\in{R}\:,\forall{n}\in{N} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)+\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+\left({n}^{\mathrm{2}} −\mathrm{1}\right){f}^{\left({n}\right)} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left(\mathrm{2}{n}+\mathrm{1}\right)} \left(\mathrm{0}\right)=\mathrm{0}\:\forall{n}\in\:{N} \\ $$

Question Number 30745 Answers: 0 Comments: 0

let U_n ={z∈C/z^n =1} simlify A_n = Σ_(α∈U_n ) (x+α)^n and B_n =Σ_(α∈ U_n ) (x−α)^n .

$${let}\:\:{U}_{{n}} =\left\{{z}\in{C}/{z}^{{n}} =\mathrm{1}\right\}\:\:{simlify} \\ $$$${A}_{{n}} =\:\sum_{\alpha\in{U}_{{n}} } \:\left({x}+\alpha\right)^{{n}} \:{and}\:{B}_{{n}} =\sum_{\alpha\in\:{U}_{{n}} } \:\:\left({x}−\alpha\right)^{{n}} . \\ $$

Question Number 30744 Answers: 0 Comments: 0

let p(x)= (x−1)^n −x^n +1 with n integr find n in ordre that p(x) have a double root.

$${let}\:{p}\left({x}\right)=\:\left({x}−\mathrm{1}\right)^{{n}} \:−{x}^{{n}} \:+\mathrm{1}\:\:{with}\:{n}\:{integr}\:{find}\:{n} \\ $$$${in}\:{ordre}\:{that}\:{p}\left({x}\right)\:{have}\:{a}\:{double}\:{root}. \\ $$

Question Number 30743 Answers: 0 Comments: 1

decompose inside R[x] p(x)=x^(2n+1) −1 then find Π_(k=1) ^n sin( ((kπ)/(2n+1))) .

$${decompose}\:{inside}\:{R}\left[{x}\right]\:\:{p}\left({x}\right)={x}^{\mathrm{2}{n}+\mathrm{1}} \:−\mathrm{1}\:{then}\:{find} \\ $$$$\prod_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\:\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)\:. \\ $$

Question Number 30742 Answers: 0 Comments: 0

prove that ∀p ∈N it exist one polynomial Q_(2p) / sin(2p+1)θ=sin^(2p+1) θ Q_(2p) (cotanθ) and degQ_(2p) =2p 2) prove that Π_(k=1) ^p tan(((kπ)/(2p+1)))=(√(2p+1)) .

$${prove}\:{that}\:\forall{p}\:\in{N}\:\:{it}\:{exist}\:{one}\:{polynomial}\:{Q}_{\mathrm{2}{p}} \:/ \\ $$$${sin}\left(\mathrm{2}{p}+\mathrm{1}\right)\theta={sin}^{\mathrm{2}{p}+\mathrm{1}} \theta\:{Q}_{\mathrm{2}{p}} \:\left({cotan}\theta\right)\:{and}\:{degQ}_{\mathrm{2}{p}} =\mathrm{2}{p} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:{tan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right)=\sqrt{\mathrm{2}{p}+\mathrm{1}}\:. \\ $$$$ \\ $$

Question Number 30741 Answers: 0 Comments: 0

let give D= R_+ ^2 −{(0,0)} and α from R let C_1 ={(x,y)∈ D/0<x^2 +y^2 ≤1 } C_2 ={(x,y) ∈D / x^2 +y^2 ≥1} study the convergence of I= ∫∫_C_1 ((dxdy)/(((√(x^2 +y^2 )) )^α )) and J=∫∫_C_2 ((dxdy)/(((√(x^2 +y^2 )) )^α )) .

$${let}\:{give}\:{D}=\:{R}_{+} ^{\mathrm{2}} \:−\left\{\left(\mathrm{0},\mathrm{0}\right)\right\}\:{and}\:\alpha\:{from}\:{R}\:{let} \\ $$$${C}_{\mathrm{1}} =\left\{\left({x},{y}\right)\in\:{D}/\mathrm{0}<{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}\:\right\} \\ $$$${C}_{\mathrm{2}} \:=\left\{\left({x},{y}\right)\:\in{D}\:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \geqslant\mathrm{1}\right\}\:{study}\:{the}\:{convergence}\:{of} \\ $$$${I}=\:\int\int_{{C}_{\mathrm{1}} } \:\:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\right)^{\alpha} }\:\:{and}\:{J}=\int\int_{{C}_{\mathrm{2}} } \:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:}\:\right)^{\alpha} }\:. \\ $$

Question Number 30738 Answers: 1 Comments: 1

A boy can swim with a speed of 26m/s in still water.He wants to swim across a 150m river from a point A to point B which is directly opposite the other side of the river.The river flows with a speed of 10m/s. i)if he always swim in the direction parallel to AB,find how far he lands downstream of B. ii)In what direction relative to the bank must he swim so as to cross directly from A to B.

$${A}\:{boy}\:{can}\:{swim}\:{with}\:{a}\:{speed}\:{of} \\ $$$$\mathrm{26}{m}/{s}\:{in}\:{still}\:{water}.{He}\:{wants}\:{to} \\ $$$${swim}\:{across}\:{a}\:\mathrm{150}{m}\:{river}\:{from} \\ $$$${a}\:{point}\:{A}\:{to}\:{point}\:{B}\:{which}\:{is}\: \\ $$$${directly}\:{opposite}\:{the}\:{other}\:{side} \\ $$$${of}\:{the}\:{river}.{The}\:{river}\:{flows}\:{with} \\ $$$${a}\:{speed}\:{of}\:\mathrm{10}{m}/{s}. \\ $$$$\left.{i}\right){if}\:{he}\:{always}\:{swim}\:{in}\:{the}\: \\ $$$${direction}\:{parallel}\:{to}\:{AB},{find}\:{how} \\ $$$${far}\:{he}\:{lands}\:{downstream}\:{of}\:{B}. \\ $$$$\left.{ii}\right){In}\:{what}\:{direction}\:{relative}\:{to} \\ $$$${the}\:{bank}\:{must}\:{he}\:{swim}\:{so}\:{as}\:{to} \\ $$$${cross}\:{directly}\:{from}\:{A}\:{to}\:{B}. \\ $$

Question Number 30737 Answers: 0 Comments: 1

∫(1/(x^2 +ln x))dx

$$\int\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{ln}\:{x}}{dx} \\ $$

Question Number 30739 Answers: 0 Comments: 0

let (u_n ) / u_1 =1−i and ∀p∈{2,3,...n} u_p =u_(p−1) j with j=e^(i((2π)/3)) 1)verify that u_1 +u_2 +u_3 =0 2)prove that ∀p∈ {4,5,...,n} u_p =u_(p−3) 3)find the value of S_n =Σ_(i=1) ^n u_i 4)calculate α_n = Σ_(p=0) ^(n−1) cos(−(π/4) +((2pπ)/3)) and β_n = Σ_(p=0) ^(n−1) sin(−(π/4) +((2pπ)/3)).

$${let}\:\left({u}_{{n}} \right)\:/\:{u}_{\mathrm{1}} =\mathrm{1}−{i}\:{and}\:\:\forall{p}\in\left\{\mathrm{2},\mathrm{3},...{n}\right\}\:{u}_{{p}} ={u}_{{p}−\mathrm{1}} {j}\:{with} \\ $$$${j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right){verify}\:{that}\:{u}_{\mathrm{1}} \:+{u}_{\mathrm{2}} \:+{u}_{\mathrm{3}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{p}\in\:\left\{\mathrm{4},\mathrm{5},...,{n}\right\}\:\:{u}_{{p}} ={u}_{{p}−\mathrm{3}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{S}_{{n}} \:=\sum_{{i}=\mathrm{1}} ^{{n}} \:{u}_{{i}} \\ $$$$\left.\mathrm{4}\right){calculate}\:\:\alpha_{{n}} =\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\left(−\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{2}{p}\pi}{\mathrm{3}}\right)\:{and} \\ $$$$\beta_{{n}} =\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{sin}\left(−\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{2}{p}\pi}{\mathrm{3}}\right). \\ $$

Question Number 30719 Answers: 1 Comments: 4

Question Number 30711 Answers: 1 Comments: 0

Question Number 33450 Answers: 1 Comments: 1

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