Theorem 64. Let {pk}1 converge to point If y # x, then {pk}1 does not converge to y. x.
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- Suppose that ∞ n = 0 anxn converges to a function y such that y'' − 2y' + y = 0 where y(0) = 0 and y'(0) = 1. Find a formula that relates an + 2, an + 1, andSuppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.Does the sequence Pn=(120e0.7n)/(4+e0.7n) converge? If so what's the limit?
- Let f(x) = (x − 3)^5 and x0 is not equal to 3. For each n ≥ 0, determine xn+1 from xn by using Newton’s method for finding the root of the equation f(x) = 0. Show that the sequence {xn} converges to 3 linearly with rate 4/5.Consider the function f(x)=1/x Show that the interval of convergence is (0, 2).Show that if (sn,tn) converges to (s,t) in ℝ2 with the usual d2 metric, then the sequence converges to the same limit in the d0 ("max") metric.