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.

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.

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

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 (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

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 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 {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

Prove that if {xn} converges to x does not equal to 0 and xn is non-zero...

Prove that if {xn} converges to x does not equal to 0 and xn is non-zero for all n, then there exists m > 0 so that |xn| ≥ m for all n.

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, ·...

Let Xn is a simple random walk (p = 1/2) on {0, 1, · · ·...

Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing boundaries. Suppose X0 = 50. Let T = min{j : Xj = 0 or N}. Let Fn denote the information contained in X1, · · · , Xn. (1) Verify that Xn is a martingale. (2) Find P(XT = 100). (3) Let Mn = X2 n − n. Verify that Mn is also a martingale. (4) It is known that...

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