Exercise 5: Let an be an increasing sequence, let bn be a decreasing seqeuence, and suppose...
please prove Let - andd On -n+1 Show that a is an increasing sequence, that bn is a decreas- Let - andd On -n+1 Show that a is an increasing sequence, that bn is a decreas-
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
Let pn = (an+bn)/2 , p = limn→∞pn, and en = p−pn. Here [an,bn], with n ≥ 1, denotes the successive intervals that arise in the Bisection method when it is applied to a continuous function f. Show that |pn −pn+1| = .(b1-a1)
show that the sequence is eventually decreasing or increasing. and determine the smallest value of n for which the term is decreasing or increasing. (b) {
Problem 3: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be a sequence such that lant - and by for all ne N. Prove that (an)nzo is a convergent sequence. (Hint: We did something similar in class before.)
Let (an)nen be a bounded sequence in R. For all n e N define bn = sup{am, On+1, On+2,...}. (You do not have to show that the supremum exists.) (a) Prove that the sequence (bn)nen is a monotone sequence. (b) Prove that the sequence (bn)nen is convergent. (c) Prove or disprove: lim an = lim bre. 100 000
Exercise 5.17. Let {en}nez be the set of trigonometric functions. Suppose that {an}nez, {bn}nez are sequences of complex numbers, f = Enez anen, and g = Enez bnen, where the equalities are in the Lº sense. Show that (f, g) = Enez anbr. In particular, show that || F || 22 (1) - Enez lan?
4. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 3n - 7 a) b) an 7n+ 5 2 пл an = COS
Determine if the next sequence is monotonous. If so, indicate if it is increasing, decreasing, not increasing or not decreasing. Www In(n+3) n + 3
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion defined by bo - 1 and bn- k-0 n E N. Show that bn-- Hint: Use a) with e*e*1 and the inverse of a power series found in the lecture. Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion...