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Question Number 204477 Answers: 1 Comments: 0

lim_(n→∞) (2n∫_0 ^1 (x^n /(1+x^2 ))dx)^n =?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\right)^{\mathrm{n}} =? \\ $$

Question Number 204472 Answers: 2 Comments: 0

Calculate ... Ω=Σ_(k=1) ^n ⌊(( 1)/( (e)^(1/k) −1)) ⌋ =?

$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Calculate}\:... \\ $$$$\:\:\:\:\:\:\:\Omega=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\lfloor\frac{\:\mathrm{1}}{\:\sqrt[{{k}}]{{e}}\:−\mathrm{1}}\:\rfloor\:=? \\ $$$$ \\ $$

Question Number 204471 Answers: 2 Comments: 0

Question Number 204469 Answers: 1 Comments: 0

Given the function f(x) = { ((2, 0< x <2)),((−2, −2 <x < 0)) :} of period 4 (a) sketch the graph of y = f(x) , for −6 < x < 6 (b) Find the Fourier coefficient a_0 , a_n , and b_n (c) write down the Fourier series. (d) hence show that Σ_(n=1) ^∞ (((−1)^n )/(2n−1)) = (π/4)

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:=\:\begin{cases}{\mathrm{2},\:\mathrm{0}<\:{x}\:<\mathrm{2}}\\{−\mathrm{2},\:−\mathrm{2}\:<{x}\:<\:\mathrm{0}}\end{cases} \\ $$$$\mathrm{of}\:\mathrm{period}\:\mathrm{4} \\ $$$$\left(\mathrm{a}\right)\:\:\mathrm{sketch}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of}\:{y}\:=\:{f}\left({x}\right)\:,\:\mathrm{for}\:−\mathrm{6}\:<\:{x}\:<\:\mathrm{6} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{Fourier}\:\mathrm{coefficient}\:{a}_{\mathrm{0}} ,\:{a}_{{n}} ,\:\mathrm{and}\:{b}_{{n}} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{write}\:\mathrm{down}\:\mathrm{the}\:\mathrm{Fourier}\:\mathrm{series}.\: \\ $$$$\left(\mathrm{d}\right)\:\mathrm{hence}\:\mathrm{show}\:\mathrm{that}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}−\mathrm{1}}\:=\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 204468 Answers: 1 Comments: 0

How Can we prove Σ_(h=−∞) ^∞ J_h (z)=1

$$\mathrm{How}\:\mathrm{Can}\:\mathrm{we}\:\mathrm{prove}\:\underset{{h}=−\infty} {\overset{\infty} {\sum}}\:{J}_{{h}} \left({z}\right)=\mathrm{1} \\ $$

Question Number 204441 Answers: 0 Comments: 2