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Question Number 73260 Answers: 1 Comments: 0

Question Number 73255 Answers: 1 Comments: 0

Question Number 73247 Answers: 1 Comments: 0

Question Number 73243 Answers: 1 Comments: 0

prove that for z ∈C arctanz =(1/(2i))ln(((1+iz)/(1−iz)))

$${prove}\:{that}\:\:{for}\:{z}\:\in{C}\:\:\:{arctanz}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}{ln}\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right) \\ $$$$ \\ $$

Question Number 73238 Answers: 1 Comments: 1

let 0<a<1 calculate ∫_0 ^∞ ((ln(t)t^(a−1) )/(1+t))dt and ∫_0 ^∞ ((ln^2 (t)t^(a−1) )/(1+t))dt

$${let}\:\mathrm{0}<{a}<\mathrm{1}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}^{\mathrm{2}} \left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt} \\ $$

Question Number 73235 Answers: 0 Comments: 0

Question Number 73261 Answers: 1 Comments: 1

calculate Σ_(n=1) ^∞ (n^4 +2n^2 −3)(x^n /(n!)) in case of convergence.

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left({n}^{\mathrm{4}} +\mathrm{2}{n}^{\mathrm{2}} −\mathrm{3}\right)\frac{{x}^{{n}} }{{n}!}\:{in}\:{case}\:{of}\:{convergence}. \\ $$

Question Number 73231 Answers: 1 Comments: 1

find the sum of Σ_(n=0) ^∞ (n^2 −3n+1)e^(−n)

$${find}\:{the}\:{sum}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \left({n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{1}\right){e}^{−{n}} \\ $$

Question Number 73230 Answers: 0 Comments: 0

calculate A_n =∫_0 ^∞ ((1+x^n )/(2+x^(2n) ))dx and J_n =∫_0 ^∞ ((2+x^(3n) )/(5+x^(7n) ))dx with n integr natural not 0

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}^{{n}} }{\mathrm{2}+{x}^{\mathrm{2}{n}} }{dx}\:\:{and}\:{J}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}+{x}^{\mathrm{3}{n}} }{\mathrm{5}+{x}^{\mathrm{7}{n}} }{dx} \\ $$$${with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0} \\ $$

Question Number 73227 Answers: 1 Comments: 0

find the polynom T_n wich verify T_n (cosθ)=cos(nθ) ∀n integr ∀θ real 1) find T_0 ,T_1 and T_2 and prove that T_(n+2) =2x T_(n+1) −T_n 3) find deg(T_n ) and T_n (1) ,T_n (−1) 4) find T^′ (cosθ) for 0<θ<π and prove that (1−x^2 )T_n ′′−xT′_n +n^2 T_n =0 5) find roots of T_n and decompose T_n inside R[x] 6) find the value of Π_(k=0) ^(n−1) cos((((2k+1)π)/(2n)))

$${find}\:{the}\:{polynom}\:{T}_{{n}} \:{wich}\:{verify}\:{T}_{{n}} \left({cos}\theta\right)={cos}\left({n}\theta\right) \\ $$$$\forall{n}\:{integr}\:\forall\theta\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{T}_{\mathrm{0}} ,{T}_{\mathrm{1}} {and}\:{T}_{\mathrm{2}} {and}\:{prove}\:{that}\: \\ $$$${T}_{{n}+\mathrm{2}} =\mathrm{2}{x}\:{T}_{{n}+\mathrm{1}} −{T}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{deg}\left({T}_{{n}} \right)\:{and}\:{T}_{{n}} \left(\mathrm{1}\right)\:,{T}_{{n}} \left(−\mathrm{1}\right) \\ $$$$\left.\mathrm{4}\right)\:{find}\:{T}\:^{'} \left({cos}\theta\right)\:{for}\:\mathrm{0}<\theta<\pi\:\:{and}\:{prove}\:{that} \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){T}_{{n}} ''−{xT}'_{{n}} \:+{n}^{\mathrm{2}} \:{T}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{roots}\:{of}\:{T}_{{n}} \:{and}\:{decompose}\:{T}_{{n}} {inside}\:{R}\left[{x}\right] \\ $$$$\left.\mathrm{6}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\left(\frac{\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{\mathrm{2}{n}}\right) \\ $$

Question Number 73225 Answers: 0 Comments: 0

calculate f(x)=∫_0 ^π ln(x^2 −2xcosθ +1)dθ with x real.

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:{x}\:{real}. \\ $$

Question Number 73224 Answers: 1 Comments: 0

find the coefficient a_k of term x^k in Π_(r=1) ^n (1+x^r ) with 0≤k≤((n(n+1))/2) example: n=100, k=50

$${find}\:{the}\:{coefficient}\:{a}_{{k}} \:{of}\:{term}\:{x}^{{k}} \:{in} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}+{x}^{{r}} \right)\:{with}\:\mathrm{0}\leqslant{k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$ \\ $$$${example}:\:{n}=\mathrm{100},\:{k}=\mathrm{50} \\ $$

Question Number 73223 Answers: 0 Comments: 5

lim (0/∞) =^(?) 0 ? lim (∞/0) =^(?) ∞ ?

$$\boldsymbol{{lim}}\:\frac{\mathrm{0}}{\infty}\:\overset{?} {=}\:\mathrm{0}\:? \\ $$$$\boldsymbol{{lim}}\:\frac{\infty}{\mathrm{0}}\:\overset{?} {=}\:\infty\:? \\ $$

Question Number 73222 Answers: 1 Comments: 0