Welcome to calculus. I'm professor Ghrist. We're about to begin lecture 53,
on absolute and conditional convergence. >> Infinite series either converge or
diverge, right? Well, the real situation is a bit
more complicated than that. In this lesson we'll focus attention
on series that have both positive and negative terms, and we'll refine our notion of convergence
into absolute and conditional. >> When it comes to determining
convergence or divergence, we have at our disposal
a collection of tests. Sometimes, those tests work well. Sometimes, well, it's not so
obvious how to proceed. Consider the following three series,
all of which look very complicated. Following the rule of
doing the ratio test first is maybe not such a good idea
in this particular case. But I claim that these series
are easy to figure out. The first two converge, the last diverges. Why is it so easy? It is easy because of this term in front,
the negative one to the n. A term which, ironically,
renders most of our prior tests useless. However, these are examples
of alternating series, series whose terms alternate
between positive and negative. Such series are very easy to work with. One often writes them in the form,
sum over n of negative one to the n a sub n,
where the a sub n terms are positive. Let's do one last test,
the alternating series test. It begins as follows. If we have a decreasing positive sequence,
a sub n, then we're going to look at the limit of
these terms just as in the nth term test. The alternating series. Sum over n negative one to
the n a sub n converges if and only if the limit is zero,
and it diverges if and only if the limit is non-zero. This is in contradistinction to the nth
term test, which was not if and only if. Now the bad part of this test
is it's not very applicable. You must have an alternating
series with decreasing terms. However, this test is as easy as can be,
and because of the ubiquity of alternating
series it's really quite useful. How can it be that we get such
a strong result for this test? Well let us consider the convergence
of an alternating series. Of course, we will look at
the sequence of partial sums. That is the sum as n goes from zero to
sum T, negative one to the n a sub n. What does the convergence
of this sequence look like? Well, if it is an alternating series,
then that means we start at a zero, we jump backwards by the amount a one, then we jump forwards by the amount a two,
backwards by the amount a three, etc. If, as per our hypotheses,
the sequence of a sub ns is decreasing, that means our
jumps get smaller and smaller. This means in the limit,
there are only two possibilities. If the limit of the a sub ns is zero,
then our jump size decreases to zero and
our series converges. If the limit of the a sub n is non-zero, then we oscillate back and
forth by that limiting jump amount. As per the nth term test,
the series does not converge. Let's see how this test works in the context of the three series
with which we began our lecture. All three of these are alternating series,
and hence, the alternating
series test applies. The first is the sum of negative
one to the n, pi to the n, log, n squared plus one over n factorial times
hyperbolic cosine of n to the five-thirds. Now looking at this, it's pretty obvious that the n
factorial term dominates all others. Hence the limit of the a sub ns is
zero and this series converges. No other work necessary. Likewise, in the second series the sum
negative one to the n of one over log of log cubed of log to the fifth of n. This one also has terms
that are going to zero. Obviously, hence it converges
without further work. The last example involving the cuperative
end of the fifth minus three n cubed plus two over the square root of n cubed
plus nine n squared plus one. Well that's not half so
complicated as it looks. We could have easily computed the leading
order terms of both of these and discerned through the nth
term test that this diverges. This is true whether or
not it's an alternating series. In all these cases, it's pretty simple. The contemplation of alternating series leads us to new definitions
involving general series. We say that a series is
absolutely convergent if not only does the sum of
the ace of ns converge, but the sum of the absolute values
of the a sub n terms also converges. We say that a series is conditionally
convergent if it is a convergent series. That is,
the sum of the a sub ns converges, but the sum of the absolute values
of the a sub ns diverges. Of course, a series that only
have positive terms in it can't be conditionally convergent. A conditionally convergent series is
one that has enough negative terms, or maybe alternating terms, so as to force a weaker sort of convergence. Both of these are in
distinction to a divergent series which does not converge at all. With these definitions there are exactly
three possibilities for a series. If you're given the sequence,
then the series, the sum of these events, might diverge. If it converges, one checks
the convergence, the absolute values. If that does not converge, then you
have a conditionally convergent series. Otherwise, you have
an absolutely convergent series. These three are mutually exclusive. Every series is one of these
three types and only one. Let's see some examples. If we take the sum from one to
infinity of negative one to the n plus one times one over n squared. This is one minus a fourth,
plus a ninth, minus one sixteenth, etc. What can we say about this? Well, it definitely converges based
on the alternating series test. When we take the absolute values,
what do we get? We get a P series with P equals two. That also converges. Therefore, this is an example of
an absolutely convergent series. On the other hand, if we take what we
might call the alternating harmonic series, negative one to the n plus one,
over n. Then by the alternating series test,
this does converge. However, when we take the absolute values
of the terms we get the harmonic series, which diverges. Therefore, this is the canonical example
of a conditionally convergent series. If we take the sum of
negative one to the n, well we've already seen that,
that is a divergent series, and taking away the negative signs is not
going to change that, not at all. In general,
if we look at an alternating p series where we take one over n to the p and
have the signs alternate on it, then instead of having convergence, it's p bigger than one, and divergence for
p less than or equal to one. In this alternating scenario
we have absolute convergence. For p strictly bigger than one and
conditional convergence for p less than or equal to one. For the remainder of this course,
we're going to go back and reconsider all of the convergences
that we have seen or implied and revisit them in
light of our new definitions. Recall, from the very beginning of this
class how we showed that log of one plus x is the alternating sum
of x to the n over n. Now, you remember and I remember,
that this only works when x Is strictly less than one in absolute value,
just like the geometric series. However, I claim that we can bend
the rules just a little bit. What happens when we substitute
x equals one into this formula? We seem to get that log of
two is one minus a half, plus a third, minus a fourth,
plus a fifth, etc. Is this True? In the past, I've asked you to trust me. But now, you don't need me because you
have enough tools at your disposal. The series on the right does converge
by the alternating series test. However, since it is
an alternating harmonic series, its convergence is conditional. It limits to the value
on the left log of two. Now recall that this series
is a little bit tricky. We have argued in the past
that by rearranging the terms and
combining in the appropriate manner, we seem to get something
that is a different value. Namely, three half log two
simply by rearranging the terms in the original series that
was presented as a mystery. Here it is presented as a warning. The real reason for that strange behavior comes from the conditional
convergence of this series. Whenever you have a conditionally
convergent series, you should be cautious. It is a dangerous situation. You are so close to having a divergent
series, you've got to be careful. On the other hand, when you have an absolutely convergent
series, you're in great shape. It is a theorem that you can rearrange the
terms in an absolutely convergent series at will, as you wish and
the sum remains constant. Absolutely convergent
series are very robust and will be very helpful when you're
working with tailor series. On the other hand, this is not the case
for conditionally convergent series. It is a result whose proof will
not fit in this margin that given a conditionally convergent series,
you can rearrange the terms to sum up to any number you wish. They are a bit dangerous, beware of
them but trust in absolute convergence. >> The distinction between absolute and conditional convergence may
seem a little academic. After all, what does it matter
to applications in the sciences? Oh, it makes a big difference,
as we'll see in our next two lessons, which will take us back to the very
beginnings of this course.