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I need some assistance with problem #38, please?
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- what is the Sn and does it diverge or converge6. Consider xn+1 = (1/3)(2xn - 9/xn2). Does it converge for any nonzero initial point? If so, to what values?A time series with a periodic component can be constructed fromxt = U1 sin(2πω0t) + U2 cos(2πω0t),where U1 and U2 are independent random variables with zero means andE(U21 ) = E(U22 ) = σ2. The constant ω0 determines the period or time ittakes the process to make one complete cycle. Show that this series is weaklystationary with autocovariance functionγ(h) = σ2 cos(2πω0h
- Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.Show that the series is DIVERGENT.Use the Cauchy Condensation test to prove that ∑ n = 2 to ∞ 1/( n (ln(n))^ p)) converges if p > 1 and diverges if p ≤ 1. (Make sure you verify that the hypothesis of the Cauchy Condensation test are met)
- Which of the following can the divergence test be used on to directly establish converges or divergesIf the Taylor series forf(x) =e3x, centered atx= 0is∞∑n=03nxnn!, what is the Taylor seriesforg(x) = 3e3xcentered atx= 0?compare the series (1+x)-3=1-3x+6x2-10x3........ with ex= 1+x +x2/2 +x3/6............ Are they both convergent? How can you prove this?