fn(x) = { n2x for 0<=x<= 1/n, -n2x + 2n for 1/n<= x<= 2/n, 0 for 2/n<= x<= 1., is uniformly convergent on [0,1].
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Donotcopy fn(x) = { n2x for 0<=x<= 1/n,
-n2x + 2n for 1/n<= x<= 2/n,
0 for 2/n<= x<= 1., is uniformly convergent on [0,1].?
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- fn(x) = { n2x for 0<=x<= 1/n, -n2x + 2n for 1/n<= x<= 2/n, 0 for 2/n<= x<= 1., is uniformly convergent on [0,1].?ASAP Donotcopy fn(x) = { n2x for 0<=x<= 1/n, -n2x + 2n for 1/n<= x<= 2/n, 0 for 2/n<= x<= 1., is uniformly convergent on [0,1].?Sequences {xn} and {yn} such that they are both convergent and {xn+yn} maps to 5
- For sequence of functions {nxe-nx} for x ∈ (0 + 1), what is the uniform norm of fn (x) - f(x) on (0 + x). is the sequence uniformly convergent?(b) A sequence (fn) of differentiable functions such that (fn ) converges uniformly but the original sequence (fn) does not converge for any x ∈ R.How to determine nln(n)/(n+1)^3 is convergent or divergent?
- Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.Theorem: Suppose (Xn)n≥1 is a sequence of random variables with corresponding momentgenerating functions M_Xn , and X is a random variable with moment generating functionM_X such that for some δ > 0 we have M_X (t) < ∞ for all t ∈ (−δ, δ). If lim n→∞ MXn (t) = MX (t) for all t, then lim n→∞ F_Xn (x) = F_X (x) for all x where F_X is continuous. That is, if the moment generating functions of X_n converge to the moment generating function of X, then the distribution of X_n converges to the distribution of X. Use this to show that if Sn ∼ Binomial(n, λ/n ), then the distribution of Sn converges to Poisson(λ) as n → ∞.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?