Let (In) and (yn) be sequences of positive numbers such that lim(xn/Yn) = +∞, show that if (n) is bounded, then lim(yn) = 0.
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- (i) Let gn(x) =1/n(1 + x2) For any fixed x ∈ R, we can see that limgn(x) = 0 so that g(x) = 0 is the pointwise limit of the sequence (gn) on R. Is this convergence uniform? The observation that 1/(1 + x2) ≤ 1 for all x ∈ R implies that .Determine the infinite limit. limit as x approaches 3 from the right side of ln(x2-9)Let x1 > 1 and xn+1 = 2 − 1 / xn for n ≥ 2. Show that ( xn ) converges and find its limit.
- Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for k = 1 to infinity converges and find its limit.(c) A sequence (fn) of differentiable functions such that both (fn) and (fn )converge uniformly but f = lim fn is not differentiable at some pointSuppose that F(u) denotes the DFT of the sequence of f(x)={1, 2, 3, 4}? What is the value of F(14)? (Hint: DFT periodicity)