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Question Number 50406 Answers: 0 Comments: 2

1) decompose at simple elements U_n = ((n x^(n−1) )/(x^n −1)) 2) calculste ∫_0 ^(2π) (dt/(x−e^(it) ))

$$\left.\mathrm{1}\right)\:{decompose}\:{at}\:{simple}\:{elements} \\ $$$${U}_{{n}} =\:\frac{{n}\:{x}^{{n}−\mathrm{1}} }{{x}^{{n}} −\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculste}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{x}−{e}^{{it}} } \\ $$

Question Number 50405 Answers: 1 Comments: 0

let V_n = (1/(2n+1)) +(1/(2n+3)) +...+(1/(4n−1)) determine lim_(n→+∞) V_n

$${let}\:\:{V}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{3}}\:+...+\frac{\mathrm{1}}{\mathrm{4}{n}−\mathrm{1}} \\ $$$${determine}\:{lim}_{{n}\rightarrow+\infty} \:{V}_{{n}} \\ $$

Question Number 50404 Answers: 0 Comments: 0

find lim_(n→+∞) U_n with U_n =(Σ_(k=1) ^(n ) ch((1/(√(k+n)))))−n

$${find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:{with} \\ $$$${U}_{{n}} =\left(\sum_{{k}=\mathrm{1}} ^{{n}\:} \:{ch}\left(\frac{\mathrm{1}}{\sqrt{{k}+{n}}}\right)\right)−{n} \\ $$$$ \\ $$

Question Number 50403 Answers: 0 Comments: 0

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

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

Question Number 50402 Answers: 0 Comments: 0

study and give the graph for f(x) = e^(−x^2 ) ln(1+3x)

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for} \\ $$$${f}\left({x}\right)\:=\:{e}^{−{x}^{\mathrm{2}} } {ln}\left(\mathrm{1}+\mathrm{3}{x}\right) \\ $$

Question Number 50401 Answers: 1 Comments: 0

study and give the graph for g(x)=(x +(1/x))^x^2

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for}\: \\ $$$${g}\left({x}\right)=\left({x}\:+\frac{\mathrm{1}}{{x}}\right)^{{x}^{\mathrm{2}} } \\ $$

Question Number 50400 Answers: 1 Comments: 0

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

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

Question Number 50399 Answers: 0 Comments: 0

find lim_(x→+∞) (ch(√(x+1))−ch(√x))^(1/(√x))

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

Question Number 50398 Answers: 0 Comments: 0

find lim_(t→+∞) ((sh((√(t^2 +t)))−sh(√(t^2 −t)))/((1+(1/t))^t^2 −(t^6 /6)ln^2 (t)))

$${find}\:{lim}_{{t}\rightarrow+\infty} \:\:\:\frac{{sh}\left(\sqrt{{t}^{\mathrm{2}} +{t}}\right)−{sh}\sqrt{{t}^{\mathrm{2}} −{t}}}{\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right)^{{t}^{\mathrm{2}} } \:\:−\frac{{t}^{\mathrm{6}} }{\mathrm{6}}{ln}^{\mathrm{2}} \left({t}\right)} \\ $$

Question Number 50397 Answers: 1 Comments: 0

calculate artan(2)+arctan(5)+arctan(8)

$${calculate}\:{artan}\left(\mathrm{2}\right)+{arctan}\left(\mathrm{5}\right)+{arctan}\left(\mathrm{8}\right) \\ $$

Question Number 50396 Answers: 0 Comments: 0

finf all functions f continues with verfy f(2x+1)=f(x) ∀x∈R

$${finf}\:{all}\:{functions}\:{f}\:{continues}\:{with}\:{verfy} \\ $$$${f}\left(\mathrm{2}{x}+\mathrm{1}\right)={f}\left({x}\right)\:\:\forall{x}\in{R} \\ $$

Question Number 50395 Answers: 0 Comments: 0

let U_n =(e−(1+(1/n))^n )^((√(n^2 +2))−(√(n^2 +1))) calculate lim_(n→+∞) U_n

$${let}\:{U}_{{n}} =\left({e}−\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \right)^{\sqrt{{n}^{\mathrm{2}} \:+\mathrm{2}}−\sqrt{{n}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$

Question Number 50394 Answers: 0 Comments: 0

let x∈]0,1[ prove that the equation tan(((πx)/2))=(π/(2nx)) have only one solution x_n 2) study tbe sequence (x_n ) and find a equivalent of x_n

$$\left.{let}\:\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\:{prove}\:{that}\:{the}\:{equation}\right. \\ $$$${tan}\left(\frac{\pi{x}}{\mathrm{2}}\right)=\frac{\pi}{\mathrm{2}{nx}}\:{have}\:{only}\:{one}\:{solution}\:{x}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{tbe}\:{sequence}\:\left({x}_{{n}} \right)\:{and}\:{find}\:{a}\:{equivalent}\:{of}\:{x}_{{n}} \\ $$

Question Number 50392 Answers: 0 Comments: 0

let u_0 =5 and ∀n∈N u_(n+1) =u_n +(1/n) prove that 45<u_(1000) <45,1

$${let}\:{u}_{\mathrm{0}} =\mathrm{5}\:{and}\:\forall{n}\in{N}\:\:\:{u}_{{n}+\mathrm{1}} ={u}_{{n}} \:+\frac{\mathrm{1}}{{n}} \\ $$$${prove}\:{that}\:\mathrm{45}<{u}_{\mathrm{1000}} <\mathrm{45},\mathrm{1} \\ $$

Question Number 50391 Answers: 0 Comments: 0

let f(t) =(t/(√(1+t))) study the sequence S_n =Σ_(k=1) ^n f((k/n^2 )).

$${let}\:{f}\left({t}\right)\:=\frac{{t}}{\sqrt{\mathrm{1}+{t}}} \\ $$$${study}\:{the}\:{sequence}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right). \\ $$

Question Number 50390 Answers: 0 Comments: 0

study the sequence u_1 =ln(2) and u_n =Σ_(k=1) ^(n−1) ln(2−u_k ) .

$${study}\:{the}\:{sequence}\:{u}_{\mathrm{1}} ={ln}\left(\mathrm{2}\right)\:{and} \\ $$$${u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} {ln}\left(\mathrm{2}−{u}_{{k}} \right)\:. \\ $$

Question Number 50388 Answers: 0 Comments: 0

find inf_((a,b)∈R^2 ) ∫_0 ^1 x^2 (ln(x)−ax−b)^2 dx

$${find}\:{inf}_{\left({a},{b}\right)\in{R}^{\mathrm{2}} } \:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} \left({ln}\left({x}\right)−{ax}−{b}\right)^{\mathrm{2}} {dx} \\ $$

Question Number 50387 Answers: 0 Comments: 0

E is a euclidian space and f from E to E verify ∀(x,y) ∈E^2 (x ∣f(y))=(f(x)∣y) prove that f is linear.

$${E}\:{is}\:{a}\:{euclidian}\:{space}\:{and}\:{f}\:\:{from}\:{E}\:{to}\:{E}\:{verify} \\ $$$$\forall\left({x},{y}\right)\:\in{E}^{\mathrm{2}} \:\:\:\:\left({x}\:\mid{f}\left({y}\right)\right)=\left({f}\left({x}\right)\mid{y}\right)\:{prove}\:{that}\:{f}\:{is}\:{linear}. \\ $$

Question Number 50386 Answers: 1 Comments: 0

let A = (((1 0 0)),((0 −2 −9)) ) (0 1 4 ) 1) calculate (A−I)^3 2) conclude A^n for n integr.

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{0}\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:−\mathrm{2}\:\:−\mathrm{9}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{4}\:\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\left({A}−{I}\right)^{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{conclude}\:\:{A}^{{n}} \:{for}\:{n}\:\:{integr}. \\ $$

Question Number 50385 Answers: 0 Comments: 1

p is a polynom having n roots simples with x_i ≠+^− 1 caculate Σ_(k=1) ^n (1/(1−x_i )) and Σ_(k=1) ^n (1/(1−x_i ^2 ))

$${p}\:{is}\:{a}\:{polynom}\:{having}\:{n}\:{roots}\:{simples}\:{with}\:{x}_{{i}} \neq\overset{−} {+}\mathrm{1} \\ $$$${caculate}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{i}} }\:\:{and}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{i}} ^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 50389 Answers: 0 Comments: 0

find the sequence u_n wich verify u_(n+2) +4u_(n+1) −4u_n =n

$${find}\:{the}\:{sequence}\:{u}_{{n}} \:\:{wich}\:{verify}\: \\ $$$${u}_{{n}+\mathrm{2}} \:+\mathrm{4}{u}_{{n}+\mathrm{1}} −\mathrm{4}{u}_{{n}} ={n} \\ $$

Question Number 50384 Answers: 1 Comments: 1

find ∫ (dx/((1−x^2 )(1−x^3 ))) 2) calculate ∫_2 ^(√5) (dx/((1−x^2 )(1−x^3 )))

$${find}\:\int\:\:\:\frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$

Question Number 50383 Answers: 0 Comments: 0

let U_n ={(x,y)∈N^2 /2x+3y=n} prove that U_n =U_(n−2) +U_(n−3) −U_(n−5) for n≥5 .

$${let}\:{U}_{{n}} =\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} /\mathrm{2}{x}+\mathrm{3}{y}={n}\right\} \\ $$$${prove}\:{that}\:{U}_{{n}} ={U}_{{n}−\mathrm{2}} \:+{U}_{{n}−\mathrm{3}} −{U}_{{n}−\mathrm{5}} \\ $$$${for}\:{n}\geqslant\mathrm{5}\:. \\ $$

Question Number 50381 Answers: 0 Comments: 0

calculate S_n =Σ_(p=1) ^n (p/(1+p^2 +p^4 )) and determine lim_(n→+∞) S_n

$${calculate}\:{S}_{{n}} =\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\frac{{p}}{\mathrm{1}+{p}^{\mathrm{2}} \:+{p}^{\mathrm{4}} } \\ $$$${and}\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 50380 Answers: 0 Comments: 0

1) decompose F(x)=(1/((x−a)^n (x−b)^n )) witha≠b 2) find the values of Σ_(k=0) ^(n−1) (C_(p+k−1) ^k /2^(k+p) ) + Σ_(k=0) ^(p−1) (C_(n+k−1) ^k /2^(k+n) )

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)=\frac{\mathrm{1}}{\left({x}−{a}\right)^{{n}} \left({x}−{b}\right)^{{n}} } \\ $$$${witha}\neq{b} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{{C}_{{p}+{k}−\mathrm{1}} ^{{k}} }{\mathrm{2}^{{k}+{p}} }\: \\ $$$$+\:\sum_{{k}=\mathrm{0}} ^{{p}−\mathrm{1}} \:\:\:\frac{{C}_{{n}+{k}−\mathrm{1}} ^{{k}} }{\mathrm{2}^{{k}+{n}} } \\ $$

Question Number 50379 Answers: 0 Comments: 0

let U_n ={z∈C /z^n =1} find two polynom A(x) and B(x) verify Σ_(w∈U_n ) (w/((x−w)^2 )) =((A(x))/(B(x)))

$${let}\:{U}_{{n}} =\left\{{z}\in{C}\:/{z}^{{n}} =\mathrm{1}\right\}\:\:{find}\:{two}\:{polynom}\:{A}\left({x}\right) \\ $$$${and}\:{B}\left({x}\right)\:{verify}\:\:\sum_{{w}\in{U}_{{n}} } \:\:\:\:\frac{{w}}{\left({x}−{w}\right)^{\mathrm{2}} }\:=\frac{{A}\left({x}\right)}{{B}\left({x}\right)} \\ $$

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