Question

prove that a sequence converges if and only if all subsequences converge to the same limit

Answer #1

Show that if sequence (an) converges, then all the rearrangement
of (an) converges, and converge to the same limit

Prove that if a sequence converges to a limit x then very
subsequence converges to x.

Determine whether the following sequences converge or diverge.
If a sequence converges, find its limit. If a sequence diverges,
explain why.
(a) an = ((-1)nn)/
(n+sqrt(n))
(b) an = (sin(3n))/(1- sqrt(n))

Prove that a sequence (un such that n>=1)
absolutely converges if the limit as n approaches infinity of
n2un=L>0

Using the definition of convergence of a sequence, prove that the
sequence converges to the proposed limit.
lim (as n goes to infinity) 1/(n^2) = 0

Prove that if a sequence is a Cauchy sequence, then it
converges.

Use L’Hopital’s Rule to determine if the following sequences
converge or diverge. If the sequence converges, what does it
converge to?
(a) an = (n^2+3n+5)/(n^2+e^n)
(b) bn = (sin(n −1 ))/( n−1)
(c) cn = ln(n)/ √n

In this task, you will write a proof to analyze the limit of a
sequence.
ASSUMPTIONS
Definition: A sequence {an} for n = 1 to ∞ converges
to a real number A if and only if for each ε > 0 there is a
positive integer N such that for all n ≥
N, |an – A| < ε .
Let P be 6. and Let Q be 24.
Define your sequence to be an = 4 +
1/(Pn +...

suppose that the sequence (sn) converges to s. prove that if s
> 0 and sn >= 0 for all n, then the sequence (sqrt(sn))
converges to sqrt(s)

Determine whether the sequence converges or diverges. If it
converges, find the limit. (If an answer does not exist, enter
DNE.)
an = (4^n+1) /
9^n

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