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Question Number 36934 Answers: 0 Comments: 0

calculate [ Σ_(k=1) ^(10^4 ) (1/(√k)) ].

$${calculate}\:\left[\:\sum_{{k}=\mathrm{1}} ^{\mathrm{10}^{\mathrm{4}} } \:\:\frac{\mathrm{1}}{\sqrt{{k}}}\:\right]. \\ $$

Question Number 36933 Answers: 0 Comments: 0

find lim_(n→+∞) Σ_(i=1) ^n Σ_(j=1) ^n (((−1)^(i+j) )/(i+j)) .

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\sum_{{i}=\mathrm{1}} ^{{n}} \:\sum_{{j}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{i}+{j}} }{{i}+{j}}\:. \\ $$

Question Number 36932 Answers: 0 Comments: 0

let f ∈ C^0 ([0,π],R) prove that lim_(n→+∞) ∫_0 ^π f(x) ∣sin(nx)∣dx =(2/π) ∫_0 ^π f(x)dx .

$${let}\:{f}\:\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],{R}\right)\:\:{prove}\:{that} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right)\:\mid{sin}\left({nx}\right)\mid{dx}\:=\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}\:. \\ $$

Question Number 36931 Answers: 0 Comments: 2

calculate ∫_0 ^(2π) (dt/(x −e^(it) ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{x}\:−{e}^{{it}} } \\ $$

Question Number 36930 Answers: 0 Comments: 0

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

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

Question Number 36929 Answers: 0 Comments: 0

study and give th graph of the function f(x)=x(1−(1/x))^(x+1) .

$${study}\:{and}\:{give}\:{th}\:{graph}\:{of}\:{the}\:{function} \\ $$$${f}\left({x}\right)={x}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)^{{x}+\mathrm{1}} . \\ $$

Question Number 36928 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 36927 Answers: 0 Comments: 0

f is a real function derivable on [0,1] /f(0)=0 and f(1)=1 prove that ∀n∈N ∃ (x_i )_(1≤i≤n) seqence of reals with x_i ≠x_j if i≠j and Σ_(k=1) ^n f^′ (x_k )=n.

$${f}\:{is}\:{a}\:{real}\:{function}\:{derivable}\:{on}\:\left[\mathrm{0},\mathrm{1}\right]\:/{f}\left(\mathrm{0}\right)=\mathrm{0}\:{and}\:{f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$$${prove}\:{that}\:\forall{n}\in{N}\:\:\exists\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \:{seqence}\:{of}\:{reals}\:{with}\:{x}_{{i}} \neq{x}_{{j}} \:{if}\:{i}\neq{j} \\ $$$${and}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}^{'} \left({x}_{{k}} \right)={n}. \\ $$

Question Number 36926 Answers: 1 Comments: 0

let f(x) = e^(−x^2 ) 1) prove that f^((n)) (x)=p_n (x)e^(−x^2 ) with p_n is a polynom 2) find a relation of recurrence between the p_n 3) calculate p_1 ,p_2 ,p_3 ,p_4

$${let}\:{f}\left({x}\right)\:=\:{e}^{−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)={p}_{{n}} \left({x}\right){e}^{−{x}^{\mathrm{2}} } \:\:{with}\:{p}_{{n}} \:{is}\:{a}\:{polynom} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{relation}\:{of}\:{recurrence}\:{between}\:{the}\:{p}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} ,{p}_{\mathrm{4}} \\ $$

Question Number 36925 Answers: 0 Comments: 1

find lim_(n→+∞) Π_(k=1) ^n (1+(k/n^2 )).

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

Question Number 36924 Answers: 0 Comments: 1

calculate lim_(n→+∞) (1/(2i)){ (1+((it)/n))^n −(1−((it)/n))^n )

$${calculate}\:\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{\mathrm{1}}{\mathrm{2}{i}}\left\{\:\left(\mathrm{1}+\frac{{it}}{{n}}\right)^{{n}} \:−\left(\mathrm{1}−\frac{{it}}{{n}}\right)^{{n}} \right) \\ $$

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)

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

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