Question

Telescoping Series. Let {an} ∞ n=0 be a sequence of real numbers converging to zero, limn→∞ an = 0. Let bn = an − an+1. Then the series X∞ n=0 bn converges.

Answer #1

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

Suppose {xn} is a sequence of real numbers that converges to
+infinity, and suppose that {bn} is a sequence of real numbers that
converges. Prove that {xn+bn} converges to +infinity.

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

If {sn} is a sequence of positive numbers converging
to a positive number L, show that √ {sn} converges to √
L.
Do not solve using calc but rather an N epsilon proff

Let (sn) ⊂ (0, +∞) be a sequence of real numbers. Prove that
liminf 1/Sn = 1 / limsup Sn

) 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: If x is a sequence of real numbers that converges to L,
then any subsequence of x converges to L.

Let (sn) be a sequence. Consider the set X consisting of real
numbers x∈R having the following property: There exists N∈N s.t.
for all n > N, sn< x. Prove that limsupsn= infX.

Problem 1 Let {an} be a decreasing and bounded
sequence. Prove that limn→∞ an exists and
equals inf{an}.

Let (a_n)∞n=1 be a sequence defined recursively by a1 = 1, a_n+1
= sqrt(3a_n) for n ≥ 1. we know that the sequence converges. Find
its limit.
Hint: You may make use of the property that lim n→∞ b_n = lim
n→∞ b_n if a sequence (b_n)∞n=1 converges to a positive real
number.

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