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

Consider the sequence: x₀=1/6 and x_n+1 = 2x_n- 3x_n² | for all natural numbers n.

Show:

a) x_n< 1/3 for all n.

b) x_n>0 for all n.

Hint. Use induction.

c) show x_n isincreasing.

d) calculate the limit.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn =...

2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn = 1}Un+1 + I{Xn 6= 1}Vn+1, n ≥ 0, where {(Un, Vn)|n ≥ 1} is an i.i.d. sequence of random variables such that Un is independent of Vn for each n ≥ 1 and U1−1 is Bernoulli(p) and V1−1 is Bernoulli(q) random variables. Show that {Xn|n ≥ 1} is a Markov chain and find its transition matrix. Also find P{Xn = 2}.

Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.

Show that among all collections with 2n-1 natural numbers in them there are exactly n numbers...

Show that among all collections with 2n-1 natural numbers in them there are exactly n numbers whose sum is divisible by n.

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

Consider the statement, “For all natural numbers n,n, if nn is prime, then nn is solitary.”...

Consider the statement, “For all natural numbers n,n, if nn is prime, then nn is solitary.” You do not need to know what solitary means for this problem, just that it is a property that some numbers have and others do not. Write the converse and the contrapositive of the statement, saying which is which. Note: the original statement claims that an implication is true for all n,n, and it is that implication that we are taking the converse and...

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

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

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

. Suppose that the sequence (xn) satisfies |xn –α| ≤ c | xn-1- α|2 for all n. Show by induction that c | xn- α| ≤ c | x0 - α|2n , and give some condition That is sufficient for the convergence of (xn) to α. Use part a) to estimate the number of iterations needed to reach accuracy |xn –α| < 10-12 in case c = 10 and |x0 –α |= 0.09.

Consider n numbers x1, x2, . . . , xn laid out on a circle and...

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero. (a) Show that, no matter what α is, the system has a solution. (b) Show that if α = 1 2 , then the system has a nontrivial solution. (c) Show...

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2...

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2 in an n x n array so that each row and column sum to the same constant.

Subjects