example of contractive sequence
Define the sequence by
(1) |
We see by induction that the radicand in (1) cannot become
negative; in fact we justify that
(2) |
for every : It’s clear when . If it is true for an , it implies that , i.e. .
As for the convergence of the sequence, which is not monotonic, one could think to show that
it is a Cauchy sequence. Unfortunately, it is almost impossible to directly express and
estimate the needed absolute value of . Fortunately,
the recursive definition (1) allows quite easily to estimate
. Then it turns out that it’s a question of a
contractive sequence, whence it is by the parent entry (http://planetmath.org/ContractiveSequence) a
Cauchy sequence.
We form the difference
where . Thus we can estimate its absolute value, by using (2):
Since , our sequence (1) is
contractive, consequently Cauchy. Therefore it converges to
a limit .