Question

Let 0 < θ < 1 and let (x_{n}) be a sequence where
|x_{n+1} − x_{n}| ≤ θ^{n }for n
= 1, 2, . . ..

a) Show that for any 1 ≤ n < m one has |x_{m} −
x_{n}| ≤ (θ^{n}/ 1-θ )*(1 − θ ^{m−n} ).

Conclude that (x_{n}) is Cauchy

b)If lim x_{n} = x* , prove the following error in
approximation (the "error in approximation" is the same as error
estimation in Taylor Theorem) in t: |x ^{∗} −
x_{n}| ≤ θ ^{n} /1 − θ , n = 1, 2, . . . .

Answer #1

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

If (xn) is a sequence of nonzero real numbers and if limn→∞ xn =
x where x does not equal zero; prove that lim n→∞ 1/ xn = 1/x

) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...

Prove that if (xn) is a sequence of real numbers,
then lim sup|xn| = 0 as n approaches infinity. if and
only if the limit of (xN) exists and xn
approaches 0.

Exercise 2.4.5: Suppose that a Cauchy sequence {xn} is such that
for every M ∈ N, there exists a k ≥ M and an n ≥ M such that xk
< 0 and xn > 0. Using simply the definition of a Cauchy
sequence and of a convergent sequence, show that the sequence
converges to 0.

Let (xn) be Cauchy in (M, d) and a ∈ M. Show that the
sequence
d(xn, a) converges in R. (Note: It is not given that
xn converges to a.
Hint: Use Reverse triangle inequality.)

Problem 1 Let X1, · · · , Xn IID∼ p(x; θ) = 1/2 (1 +θx), −1 <
x < 1, −1 < θ < 1. 1. Estimate θ using the method of
moments. 2. Show that the above MoM is consistent by showing it’s
mean square error converges to 0 as n goes to infinity. 3. Find its
asymptotic distribution.

Let
( xn) and (yn) be sequence with xn converge to x and yn converge to
y. prove that for dn=((xn-x)^2+(yn-y)^2)^(1/2), dn converge to
0.

Let {xn} be a non-decreasing sequence and assume that xn goes to
x as n goes to infinity. Show that for all, n in N (naturals), xn
< x. Formulate and prove an analogous result for a
non-increasing sequences.

Let xn be a sequence such that for every m ∈ N, m ≥ 2 the
sequence limn→∞ xmn = L. Prove or provide a counterexample: limn→∞
xn = L.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 54 seconds ago

asked 5 minutes ago

asked 14 minutes ago

asked 30 minutes ago

asked 31 minutes ago

asked 42 minutes ago

asked 44 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago