>> Let's take a look now at the idea of determining
convergence or divergence.
And what I'm going to do for each of these
is simply take a look at the idea of the limit,
now that we understand some properties of
taking limits of sequences, we can apply that
to determine convergence and divergence.
Now, one of the things you may be asking yourself
is, why would you really care?
Well, as you recall, earlier, I basically
defined, or quickly defined the idea of an
infinite series as the summation of an infinite
sequence.
So one of the very important aspects of an
infinite sequence is determining whether they
converge or diverge.
And we will connect the idea of a sequence,
an infinite sequence, with that of a series,
and I will actually alter show that if the
sequence diverges, then the series must diverge
also.
And if the sequence converges, then there's
a possibility that the series will converge.
So this idea of determining whether or not
a sequence diverges is actually extremely
important when working with series.
So, what I'm going to do is take a look at
this limit here.
I want the limit as n goes to infinity of
1 plus negative 1 to the n, which equals 1
plus the limit as n goes to infinity of negative
1 to the n.
This is clearly divergent.
This itself does not exist, because as we
bounce it back and forth between positive
1 and negative 1, positive 1 and negative
1.
Therefore, what I have is 1 plus something
that doesn't exist, which does not exist,
therefore An is divergent.
And again just as a reminder, "divergent"
does not mean it goes to infinity.
We very commonly, with continuous functions,
think of a divergence as going to infinity
plus or minus infinity, but that's not always
the case.
And in this particular situation, we're not
going to plus or minus infinity, we're just
bouncing back and forth this component, between
positive and negative 1, so it never settles
down.
Here, if I take a look at the limit as n goes
to infinity of 3 minus 1 over 2 to the n,
I did this in an earlier example.
This equals 3 minus the limit as n goes to
infinity of 1 over 2 to the n, this goes as
zero, so this is actually equal to 3 minus
0, which is 3, therefore convergent.
Now, I did this earlier.
I'm just repeating it because now we've actually
much more formalized the idea of what the
limit of a sequence actually is.
And actually this next example I believe I
also did earlier.
Take a look at the limit as n goes to infinity,
n over negative 3n plus 1.
Now, this goes to infinity over infinity -- well,
infinity over negative infinity -- so I can
use L'Hopital's rule.
So this converges to a negative one-third,
therefore An converges.
Or, we can say that An is convergent.
Now, I just used L'Hopital's rule here, so
let's just go ahead and work with this and
we'll formalize this a little bit more.
We can use L'Hopital's rule -- I've been using
it.
We can use L'Hopital's rule to help us with
these limits.
Here's another example using L'Hopital's rule.
Just as in this previous one, here this actually
would have gone to infinity over negative
infinity, which is a negative infinity over
infinity which is not equal to negative 1
-- don't be canceling out infinities, that's
not a good idea!
Here's another example where I can apply L'Hopital's
rule.
See, as N goes to infinity, lim of n squared
goes to infinity, as does n go to infinity,
so this thing goes to infinity over infinity,
which is an indeterminate form.
So I can say based on L'Hopital's rule, that
this is equal to the limit as n goes to infinity
of 1 over n squared times the derivative of
n squared, which is -- that's a square right
there -- 2n, divided by the derivative of
this, which is 1.
Well, this n would cancel here, so what I
end up with is the limit as n goes to infinity
of 2 over n, that will go to zero.
Therefore, An -- there we go!
-- An converges to zero.
What that's actually telling us, if we think
about this, that obviously lim of n squared
goes to infinity.
And n goes to infinity, but the mere fact
that it converges to zero, what that's really
telling us is that n goes to infinity faster
than lim of n squared, hence the reason we
end up converging to zero....
