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Question Number 30487 Answers: 0 Comments: 0

for n≥2 let x_n = ((Σ_(k=1) ^n [lnk])/(ln(n!))) find lim_(n→∞) x_n .

$${for}\:{n}\geqslant\mathrm{2}\:{let}\:\:{x}_{{n}} =\:\:\frac{\sum_{{k}=\mathrm{1}} ^{{n}} \:\left[{lnk}\right]}{{ln}\left({n}!\right)}\:\:{find}\:{lim}_{{n}\rightarrow\infty} {x}_{{n}} \:\:. \\ $$

Question Number 30486 Answers: 0 Comments: 0

let give a_n = Π_(k=1) ^n cos( (π/((k+2)!))) and b_n =Π_(k=1) ^n sin((π/((k+2)!))) 1) find a equivalent foe a_n and b_n 2) find a equivalent of a_n .b_n .

$${let}\:{give}\:{a}_{{n}} =\:\prod_{{k}=\mathrm{1}} ^{{n}} \:{cos}\left(\:\frac{\pi}{\left({k}+\mathrm{2}\right)!}\right)\:{and} \\ $$$${b}_{{n}} =\prod_{{k}=\mathrm{1}} ^{{n}} \:\:{sin}\left(\frac{\pi}{\left({k}+\mathrm{2}\right)!}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{foe}\:{a}_{{n}} \:{and}\:{b}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{a}_{{n}} .{b}_{{n}} . \\ $$

Question Number 30485 Answers: 0 Comments: 1

1) prove that ∀ x>0 (x/(x+1)) ≤ ln(1+x)≤x 2) let put S_n = Π_(k=1) ^n (1 + (k/n)) find lim_(n→∞) S_n .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{x}>\mathrm{0}\:\:\:\frac{{x}}{{x}+\mathrm{1}}\:\leqslant\:{ln}\left(\mathrm{1}+{x}\right)\leqslant{x} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{put}\:\:{S}_{{n}} =\:\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}\:+\:\frac{{k}}{{n}}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{S}_{{n}} \:\:. \\ $$

Question Number 30484 Answers: 0 Comments: 0

1) prove that if f is decreasing function we have ∫_n ^(n+1) f(t)dt <f(n)< ∫_(n−1) ^n f(t) dt . 2) let put S_n = Σ_(k=1) ^n^2 (1/(2(√k))) .calculate [S_n ].

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{if}\:{f}\:{is}\:{decreasing}\:{function}\:{we}\:{have} \\ $$$$\:\int_{{n}} ^{{n}+\mathrm{1}} {f}\left({t}\right){dt}\:<{f}\left({n}\right)<\:\int_{{n}−\mathrm{1}} ^{{n}} \:{f}\left({t}\right)\:{dt}\:\:. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{put}\:\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}^{\mathrm{2}} } \:\:\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{{k}}}\:.{calculate}\:\left[{S}_{{n}} \right]. \\ $$

Question Number 30483 Answers: 0 Comments: 0

we define the bernoulli polynomial B_n by b_0 =1 and ∀n∈ N^★ b_n ^′ =n b_(n−1) and ∫_0 ^1 b_n (t)dt=0 1) find b_n (1)−b_n (0) for n≥2 2) prove that b_n (x)=(−1)^n b_n (1−x)∀n∈N 3)calculate b_0 , b_1 ,b_2 ,b_3

$${we}\:{define}\:{the}\:{bernoulli}\:{polynomial}\:{B}_{{n}} \:{by} \\ $$$${b}_{\mathrm{0}} =\mathrm{1}\:{and}\:\forall{n}\in\:{N}^{\bigstar} \:\:\:{b}_{{n}} ^{'} ={n}\:{b}_{{n}−\mathrm{1}} \:\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {b}_{{n}} \left({t}\right){dt}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{b}_{{n}} \left(\mathrm{1}\right)−{b}_{{n}} \left(\mathrm{0}\right)\:{for}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:\:{prove}\:{that}\:{b}_{{n}} \left({x}\right)=\left(−\mathrm{1}\right)^{{n}} {b}_{{n}} \left(\mathrm{1}−{x}\right)\forall{n}\in{N} \\ $$$$\left.\mathrm{3}\right){calculate}\:{b}_{\mathrm{0}} ,\:{b}_{\mathrm{1}} ,{b}_{\mathrm{2}} \:,{b}_{\mathrm{3}} \\ $$

Question Number 30482 Answers: 0 Comments: 0

find S= Σ_(p≥1,q≥1 and D^ (p,q)=1) (1/(p^2 q^2 )) .

$${find}\:\:{S}=\:\sum_{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}\:{and}\:\hat {{D}}\left({p},{q}\right)=\mathrm{1}} \:\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} {q}^{\mathrm{2}} }\:. \\ $$

Question Number 30481 Answers: 0 Comments: 0

find the value of s_1 = Σ_(p≥1,q≥1) (1/(p^2 q^2 )) and s_2 = Σ_(p≥1,q≥1 ,pdivide q) (1/(p^2 q^2 )) .

$${find}\:{the}\:{value}\:{of}\:\:{s}_{\mathrm{1}} =\:\sum_{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}} \:\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} {q}^{\mathrm{2}} }\:\:\:{and} \\ $$$${s}_{\mathrm{2}} =\:\:\sum_{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}\:,{pdivide}\:{q}} \:\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} {q}^{\mathrm{2}} }\:. \\ $$

Question Number 30480 Answers: 0 Comments: 0

let f(x)= Σ_(k=2) ^∞ (((−1)^k )/(x+k)) 1) find D_f 2)let put δ(x)= Σ_(n=1) ^∞ (((−1)^n )/n^x ) (Rieman alternate serie) find f(x) interms of δ(x).

$${let}\:{f}\left({x}\right)=\:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{x}+{k}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right){let}\:{put}\:\delta\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} }\:\:\left({Rieman}\:{alternate}\:{serie}\right) \\ $$$${find}\:{f}\left({x}\right)\:{interms}\:{of}\:\delta\left({x}\right). \\ $$

Question Number 30479 Answers: 0 Comments: 0

integrate y^(′′) −2y^′ +y =t with y(0)=1