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

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

1. If L< 1 then {x_n} is convergent to zero

2. If L> 1 then {x_n} is divergent

Expert Solution

Colby Messinger answered 2 years ago

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.

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn ≤ lim inf(xn + yn) ≤ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. Give example where all inequalities are strict. b)Let (zn) be the sequence defined recursively by z1 = z2 = 1, zn+2 = √ zn+1 + √ zn, n = 1, 2, . . . . Prove that (zn) is convergent and find its limit. Hint; argue...

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.

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.

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 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

Please prove the following formally and clearly: Let X1 = 1. Define Xn+1 = sqrt(3 +...

Please prove the following formally and clearly: Let X1 = 1. Define Xn+1 = sqrt(3 + Xn). Show that (Xn) is convergent and find its limit.

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >...

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m > 1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the sequence is an integer plus or minus 1/3 ). Show that the sequence sn is eventually constant, i.e. after a point all terms of the sequence are the same

4. (Reflected random walk) Let {Xn|n ≥ 0} be as in Q6. Show that Xn+1 =...

4. (Reflected random walk) Let {Xn|n ≥ 0} be as in Q6. Show that Xn+1 = X0 + Zn+1 − Xn m=0 min{0, Xm + Vm+1 − Um+1}, where Zn = Xn m=1 (Vm − Um), n ≥ 1. Q5. (Extreme value process) Let {In|n ≥ 0} be an i.i.d. sequence of Z-valued random variables such that P{I1 = k} = pk, k ∈ Z and pk > 0 for some k > 0. Define Xn = max{I1, I2, ·...

Question