Suppose (Sn) is a sequence such that, for all n 1, we have 1 Sn+1 – snl < n2 Show that (sn) converges
Let S = {a, b}. Show that the language L = {w EX : na(w)<n(w) } is not regular.
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the series In ancos(nx) converges uniformly on [0, 27). This means, the partial sums Sn(x) = ) ancos(nx) define a sequence of functions {sn} = that converges uniformly on [0, 271]. Hint: First show that the sequence is Cauchy with respect to || · ||00.
Problem 5.4 (10 points) Let (Sn)n-01. be a simple, symmetric random walk with starting value So-s e R. (a) Show that ES for alln0 b) Show that ElSn+1 Sn] Sn for 0. (c)Suppose that (Sn)n-0,12,. . denotes the profit and loss from $1 bets of a gambler with initial capital So-s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)? Problem 5.4...
6. Let si = 4 and sn +1 (sn +-) for n > 0. Prove lim n→oo sn exists and find limn-oo Sn. (Hint: First use induction to show sn 2 2 and the.show (sn) is decreasing)
bn converges 18. Let (an)n=1 and (bn)n=1 be sequences in R. Show that if and lan – an+1 < oo, then anbr converges.
Find the imin and limsup oP the followi uences denoted lou Sn n+1 (b) sn = n (1+(-4)^) + n.)((-1) ) (c) sn=봬 , where ynis - bounded (d) sn =n2. sequence. Find the imin and limsup oP the followi uences denoted lou Sn n+1 (b) sn = n (1+(-4)^) + n.)((-1) ) (c) sn=봬 , where ynis - bounded (d) sn =n2. sequence.
Let S be the set of all Cauchy sequences (sn) such that sn є Q for all n. Prove that the following is an equivalence relation on the set S: (%) ~ (h) if and only if (sn tn) converges to zero. Let R denote the set of equivalence classes of S under ~
Exercise 5.22. Let (Xn)nel be a sequence of i.i.d. Poisson(a) RVs. Let Sn-X1++Xn (i) Let Zn-(Sn-nA)/Vm. Show that as n-, oo, Zn converges to the standard normal RV Z ~ N(0,1) in distribution (ii) Conclude that if Yn~Poisson(nX), then ii) Fromii) deduce that we have the following approximation which becomes more accurate as noo.
Use S - SN<bN+1 to find the smallest value of N such that Sn approximates the value of the sum S to within an error of at most 104 S = (-1)"+1 11 n(n+1)(n+2) (Give your answer as a whole number.)