. Suppose that the sequence (xn) satisfies |xn –α| ≤ c | xn-1- α|2 for all...

. Suppose that the sequence (x_n) satisfies

|x_n –α| ≤ c | x_n-1- α|² for all n.

Show by induction that c | x_n- α| ≤ c | x₀ - α|²ⁿ , and give some condition

That is sufficient for the convergence of (x_n) to α.

Use part a) to estimate the number of iterations needed to reach accuracy

|x_n –α| < 10^-12 in case c = 10 and |x₀ –α |= 0.09.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let (Xn) be a monotone sequence. Suppose that (Xn) has a Cauchy subsequence. Prove that (Xn)...

Let (Xn) be a monotone sequence. Suppose that (Xn) has a Cauchy subsequence. Prove that (Xn) converges.

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is summable.

a real sequence xn is defined inductively by x1 =1 and xn+1 = sqrt(xn +6) for...

a real sequence xn is defined inductively by x1 =1 and xn+1 = sqrt(xn +6) for every n belongs to N a) prove by induction that xn is increasing and xn <3 for every n belongs to N b) deduce that xn converges and find its limit

Let (xn) be a sequence with positive terms. (a) Prove the following: lim inf xn+1/ xn...

Let (xn) be a sequence with positive terms. (a) Prove the following: lim inf xn+1/ xn ≤ lim inf n√ xn ≤ lim sup n√xn ≤ lim sup xn+1/ xn . (b) Give example of (xn) where all above inequalities are strict. Hint; you may consider the following sequence xn = 2n if n even and xn = 1 if n odd.

Prove the following test: Let {xn} be a sequence and lim |Xn| ^1/n = L 1....

Prove the following test: Let {xn} be a sequence and lim |Xn| ^1/n = L 1. If L< 1 then {xn} is convergent to zero 2. If L> 1 then {xn} is divergent

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0...

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0 (a) Assume both α and β are unknown, find their momthod of moment estimators: αˆMOM and βˆMOM. (b) Assume α is known and β is unknown, find the maximum likelihood estimation for β. 2.

Consider the sequence: x0=1/6 and xn+1 = 2xn- 3xn2 | for all natural numbers n. Show:...

Consider the sequence: x0=1/6 and xn+1 = 2xn- 3xn2 | for all natural numbers n. Show: a) xn< 1/3 for all n. b) xn>0 for all n. Hint. Use induction. c) show xn isincreasing. d) calculate the limit.

Let a sequence {xn} from n=1 to infinity satisfy x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ...... 1. Prove...

Let a sequence {xn} from n=1 to infinity satisfy x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ...... 1. Prove that a<=xn<=b for all n>=1 2. Show |x_(n+1) - xn| <= sqrt(b)/(sqrt(a)+sqrt(b)) * |xn - x_(n-1)| for n=2,3,..... 3. Prove {xn} is a cauchy sequence and hence is convergent Please show full working for 1,2 and 3.

Question 1 - Infinite Sequences. (a). Determine an infinite sequence that satisfies the following . ....

Question 1 - Infinite Sequences. (a). Determine an infinite sequence that satisfies the following . . . (i) An infinite sequence that is bounded below, decreasing, and convergent (ii) An infinite sequence that is bounded above and divergent (iii) An infinite sequence that is monotonic and converges to 1 as n → ∞ (iv) An infinite sequence that is neither increasing nor decreasing and converges to 0 as n → ∞ (b). Given the recurrence relation an = an−1 +...

Let x1 > 1 and xn+1 := 2−1/xn for n ∈ N. Show that xn is...

Let x1 > 1 and xn+1 := 2−1/xn for n ∈ N. Show that xn is bounded and monotone. Find the limit. Prove by induction

Subjects