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Question Number 36923 Answers: 0 Comments: 1

for t≥0 and f(t)= (t/(√(1+t))) let S_n =Σ_(k=1) ^n f((k/n^2 )) study the convergence of S_n .

$${for}\:{t}\geqslant\mathrm{0}\:{and}\:\:{f}\left({t}\right)=\:\frac{{t}}{\sqrt{\mathrm{1}+{t}}}\:\:{let} \\ $$$${S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:\:{study}\:{the}\:{convergence}\:{of}\:{S}_{{n}} \:\:. \\ $$

Question Number 36922 Answers: 0 Comments: 1

(u_n )is a sequence and lim_(n→+∞) u_n =l let v_n = (1/2^n ) Σ_(k=0) ^n C_n ^k u_k prove that v_n →l(n→+∞ )

$$\left({u}_{{n}} \right){is}\:{a}\:{sequence}\:{and}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} ={l}\:{let} \\ $$$${v}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\:\sum_{{k}=\mathrm{0}} ^{{n}} {C}_{{n}} ^{{k}} \:{u}_{{k}} \:\:{prove}\:{that}\:{v}_{{n}} \:\rightarrow{l}\left({n}\rightarrow+\infty\:\right) \\ $$

Question Number 36921 Answers: 0 Comments: 0

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

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

Question Number 36920 Answers: 0 Comments: 1

let α from R and u_n −2cos(α)u_(n−1) +u_(n−2) =0 withn≥2 find u_n and study its convrgence.

$${let}\:\alpha\:{from}\:{R}\:{and}\:\:{u}_{{n}} \:−\mathrm{2}{cos}\left(\alpha\right){u}_{{n}−\mathrm{1}} \:+{u}_{{n}−\mathrm{2}} =\mathrm{0}\:\:\:{withn}\geqslant\mathrm{2} \\ $$$${find}\:{u}_{{n}} \:{and}\:{study}\:{its}\:{convrgence}. \\ $$

Question Number 36919 Answers: 0 Comments: 1

calculate f(α)= ∫_(−∞) ^(+∞) (1+αi)^(−x^2 ) dx .

$${calculate}\:{f}\left(\alpha\right)=\:\int_{−\infty} ^{+\infty} \:\left(\mathrm{1}+\alpha{i}\right)^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 36918 Answers: 0 Comments: 0

calculate ∫_0 ^(+∞) (1−i)^(−x^2 ) dx

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\left(\mathrm{1}−{i}\right)^{−{x}^{\mathrm{2}} } {dx}\: \\ $$

Question Number 36917 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) (1+i)^(−x^2 ) dx

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \left(\mathrm{1}+{i}\right)^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 36916 Answers: 0 Comments: 1

let z=r e^(iθ) fins f(z) = ∫_(−∞) ^(+∞) z^(−x^2 ) dx

$${let}\:{z}={r}\:{e}^{{i}\theta} \:\:\:\:{fins}\:{f}\left({z}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:{z}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 36915 Answers: 0 Comments: 1

let z =a+ib find f(z) = ∫_(−∞) ^(+∞) z^(−x^2 ) dx

$${let}\:{z}\:={a}+{ib}\:\:\:{find}\:\:{f}\left({z}\right)\:=\:\int_{−\infty} ^{+\infty} \:{z}^{−{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 36912 Answers: 0 Comments: 1

let ⟨p,q⟩= ∫_(−1) ^1 p(x)q(x)dx with p and q are two polynoms fromR[x] 1)let p(x)=x^n calculate ⟨p,p⟩ 2)let p(x)=1+x+x^2 +....+x^n find ⟨p,p⟩.

$${let}\:\:\langle{p},{q}\rangle=\:\int_{−\mathrm{1}} ^{\mathrm{1}} {p}\left({x}\right){q}\left({x}\right){dx}\:\:{with}\:{p}\:{and}\:{q}\:{are} \\ $$$${two}\:{polynoms}\:{fromR}\left[{x}\right] \\ $$$$\left.\mathrm{1}\right){let}\:{p}\left({x}\right)={x}^{{n}} \:\:\:{calculate}\:\langle{p},{p}\rangle \\ $$$$\left.\mathrm{2}\right){let}\:{p}\left({x}\right)=\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+....+{x}^{{n}} \\ $$$${find}\:\langle{p},{p}\rangle. \\ $$

Question Number 36911 Answers: 0 Comments: 1

p is a polynome having nroots simples x_i (1≤x_i ≤n ) with x_i ^2 ≠1 calculste Σ_(k=1) ^n (1/(1−x_k )) .

$${p}\:{is}\:{a}\:{polynome}\:{having}\:{nroots}\:{simples} \\ $$$${x}_{{i}} \:\left(\mathrm{1}\leqslant{x}_{{i}} \leqslant{n}\:\right)\:{with}\:{x}_{{i}} ^{\mathrm{2}} \:\neq\mathrm{1}\:\:{calculste} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{k}} }\:. \\ $$

Question Number 36910 Answers: 0 Comments: 0

1) decompose inside R(x) the fraction F(x)= (1/((1−x^2 )(1−x^3 ))) 2) find ∫ F(x)dx .

$$\left.\mathrm{1}\right)\:{decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int\:{F}\left({x}\right){dx}\:. \\ $$

Question Number 36909 Answers: 0 Comments: 0

let p(x)=x^3 −2x^2 −1 and α is root of p(x) prove that α∉ Q .

$${let}\:{p}\left({x}\right)={x}^{\mathrm{3}} \:−\mathrm{2}{x}^{\mathrm{2}} \:−\mathrm{1}\:{and}\:\alpha\:{is}\:{root}\:{of}\:{p}\left({x}\right) \\ $$$${prove}\:{that}\:\alpha\notin\:{Q}\:. \\ $$

Question Number 36908 Answers: 0 Comments: 0

calculate S_n = Σ_(p=1) ^n (p/(1+p +p^2 )) 2) find lim_(n→+) S_n .

$${calculate}\:{S}_{{n}} =\:\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\frac{{p}}{\mathrm{1}+{p}\:+{p}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+} \:{S}_{{n}} \:\:. \\ $$

Question Number 36907 Answers: 0 Comments: 0

let f(x)= (1/(cosx)) find f^((n)) (x)

$${let}\:\:{f}\left({x}\right)=\:\:\frac{\mathrm{1}}{{cosx}}\:\:{find}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$

Question Number 36969 Answers: 1 Comments: 0

[lim_(n→∞) (2.2^3 .2^5 .....2^(n−1) .3^2 .3^4 .....3^n )^(1/(n^2 +1)) ]^4 =?

$$\left[\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{2}.\mathrm{2}^{\mathrm{3}} .\mathrm{2}^{\mathrm{5}} .....\mathrm{2}^{\mathrm{n}−\mathrm{1}} .\mathrm{3}^{\mathrm{2}} .\mathrm{3}^{\mathrm{4}} .....\mathrm{3}^{\mathrm{n}} \right)^{\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}} \right]^{\mathrm{4}} =? \\ $$

Question Number 36905 Answers: 0 Comments: 0

p is apolynom with n roots differents let Q = p^2 +p^′ let α the number of roots of Q prove that n−1≤α≤n+1 .

$${p}\:{is}\:{apolynom}\:{with}\:{n}\:{roots}\:{differents} \\ $$$${let}\:{Q}\:=\:{p}^{\mathrm{2}} \:+{p}^{'} \:\:\:\:{let}\:\alpha\:{the}\:{number}\:{of}\:{roots}\:{of} \\ $$$${Q}\:{prove}\:{that}\:\:\:{n}−\mathrm{1}\leqslant\alpha\leqslant{n}+\mathrm{1}\:. \\ $$

Question Number 36904 Answers: 0 Comments: 1

1)decompose inside C[x] p(x)=x^(2n) −2(cosα)x^n +1 2) decopose p(x)inside R[x]

$$\left.\mathrm{1}\right){decompose}\:{inside}\:{C}\left[{x}\right] \\ $$$${p}\left({x}\right)={x}^{\mathrm{2}{n}} \:−\mathrm{2}\left({cos}\alpha\right){x}^{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{decopose}\:{p}\left({x}\right){inside}\:{R}\left[{x}\right] \\ $$

Question Number 36903 Answers: 0 Comments: 0

prove that 2^(n+1) divide [(1+(√3))^(2n+1) ] [x] mean integr part of x

$${prove}\:{that}\:\:\mathrm{2}^{{n}+\mathrm{1}} \:{divide}\:\left[\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}+\mathrm{1}} \right]\: \\ $$$$\left[{x}\right]\:{mean}\:{integr}\:{part}\:{of}\:{x} \\ $$

Question Number 36892 Answers: 1 Comments: 1

2. ∫[(√((1−x^2 )/(1+x^2 )))]dx=?

$$\mathrm{2}.\:\int\left[\sqrt{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)/\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\right]{dx}=? \\ $$

Question Number 36886 Answers: 0 Comments: 1

Question Number 36884 Answers: 0 Comments: 0

1. What will be the equation of the cueved line which is made by a fixed point in the boundary of a moving circular object in respect to an another fiexd point on the way of moving?

$$\mathrm{1}.\:\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{cueved}\:\mathrm{line}\:\mathrm{which}\:\mathrm{is}\:\mathrm{made}\:\mathrm{by}\:\mathrm{a} \\ $$$$\mathrm{fixed}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{boundary}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{moving}\:\mathrm{circular}\:\mathrm{object}\:\mathrm{in}\:\mathrm{respect}\:\mathrm{to} \\ $$$$\mathrm{an}\:\mathrm{another}\:\mathrm{fiexd}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{way}\:\mathrm{of} \\ $$$$\mathrm{moving}? \\ $$

Question Number 36880 Answers: 1 Comments: 3

Question Number 36877 Answers: 0 Comments: 0

Question Number 36876 Answers: 0 Comments: 0

Question Number 36874 Answers: 0 Comments: 0

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