Question

Define a sequence (xn)_{n≥1} recursively by x1 = 1 and
xn = 1 + 1 /(xn−1) for n > 0. Prove that limn→∞ xn = x exists
and find its value.

Answer #1

Define a sequence (xn)n≥1 recursively by x1 = 1, x2 = 2, and xn
= ((xn−1)+(xn−2))/ 2 for n > 2. Prove that limn→∞ xn = x exists
and find its value.

Consider a sequence defined recursively as X0=
1,X1= 3, and Xn=Xn-1+
3Xn-2 for n ≥ 2. Prove that Xn=O(2.4^n) and
Xn = Ω(2.3^n).
Hint:First, prove by induction that 1/2*(2.3^n) ≤ Xn
≤ 2.8^n for all n ≥ 0
Find claim, base case and inductive step. Please show step and
explain all work and details

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

) 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...

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.

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

Consider the sequence (xn)n given by x1 = 2, x2 = 2 and xn+1 =
2(xn + xn−1).
(a) Let u, w be the solutions of the equation x 2 −2x−2 = 0, so
that x 2 −2x−2 = (x−u)(x−w). Show that u + w = 2 and uw = −2.
(b) Possibly using (a) to aid your calculations, show that xn =
u^n + w^n .

Given that xn is a sequence of real numbers. If (xn) is a
convergent sequence prove that (xn) is bounded. That is, show that
there exists C > 0 such that |xn| less than or equal to C for
all n in N.

Let n ≥ 2 and x1, x2, ..., xn > 0 be such that x1 + x2 + · ·
· + xn = 1. Prove that √ x1 + √ x2 + · · · + √ xn /√ n − 1 ≤ x1/ √
1 − x1 + x2/ √ 1 − x2 + · · · + xn/ √ 1 − xn

Prove the following clearly, neatly and step-by-step:
Let X1=1. Define Xn+1 =
sqrt(3+Xn). Show that (Xn) is convergent (by
using delta/epsilon proof) and then find its limit.

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