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.

Expert Solution

Colby Messinger answered 1 year ago

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 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 (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 (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.

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 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.

convergent or divergent infinity sigma n = 1 sqrt(n^5+ n^3 -7) / (n^3-n^2+n)

integrate from infinity to 1: (1+(sin^2(x))/sqrt (x)) dx

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

Question