So we looked at the formal definition of what it means
for a sequence to converge, but as a practical matter, how
can we tell whether a given sequence converges or not?
There are two criteria that are the most commonly used for
that purpose, and it's useful to be aware of them.
The first one deals with the case where we have a sequence
of numbers that keep increasing, or at least, they
do not go down.
In that case, those numbers may go up forever without any
bound, so if you look at any particular value, there's
going to be a time at which the sequence has
exceeded that value.
In that case, we say that the sequence
converges to infinity.
But if this is not the case, if it does not converge to
infinity, which means that the entries of the
sequence are bounded--
they do not grow arbitrarily large--
then, in that case, it is guaranteed that the sequence
will converge to a certain number.
This is not something that we will attempt to prove, but it
is a useful fact to know.
Another way of establishing convergence is to derive some
bound on the distance or our sequence from the number that
we suspect to be the limit.
If that distance becomes smaller and smaller, if we can
manage to bound that distance by a certain number and that
number goes down to 0, then it is guaranteed that since this
distance goes down to 0, that the sequence, ai,
converges to a.
And there's a variation of this argument, which is the
so-called sandwich argument, and it goes as follows.
If we have a certain sequence that converges to some number,
a, and we have another sequence that converges to
that same number, a, and our sequence is somewhere
in-between, then our sequence must also converge to that
particular number, a.
So these are the usual ways of quickly saying something about
the convergence of a given sequence, and we will be often
using this type of argument in this class, but without making
a big fuss about them, or without even referring to
these facts in an explicit manner.