If a series has
nothing but positive terms,
we know how to deal with it.
We can use the Integral Test on it,
we can use Comparison Test on it,
we can use a Limit Comparison Test on it.
If the terms alternate
between positive and negative,
then we have some tools
for dealing with it.
But if a series has a
bunch of positive terms
and a bunch of negative terms
without any special pattern
about positive or negative,
then you wonder what to do.
And the trick is to look
at the absolute values.
The absolute values of the terms--
well by definition they're all positive.
And if the sum of the
absolute values converges,
then we say that the original
sequence converges absolutely.
Absolutely in this case doesn't
mean "positively sure 100%."
It means absolute value.
Now, the big deal about
absolute convergence is that
if it converges absolutely,
then it converges.
So, for instance,
the reason for that is
you look at just the
positive part of the numbers.
If you look at an plus
the absolute value of an,
well if an is negative,
then that adds up to zero.
If an is positive, you get
twice the absolute value.
And,
so an plus the absolute value
is somewhere between zero
and twice the absolute value.
Actually it's one or 
the other of those, but
it's somewhere in between.
And from that,
we say, suppose that the sum
of the absolute values converges.
Well then,
the sum of twice the absolute
values have to converge,
because that's just multiplying by two.
But then the sum of an plus
the absolute values converge
by the Comparison Test.
If you're smaller than
a series that converges,
and you're a positive series,
then you have to converge.
And then an plus the absolute value
minus the absolute value has to converge
because this part converges,
that's a convergent series,
and this is a convergent series.
And the difference of
two convergent series
is a convergent series.
And of course that's just an itself.
So, if it converges absolutely,
it converges.
Now,
I gave you a proof,
but you might still be wondering,
"Yeah, but why does it work again?"
And the idea is that, if
it converges absolutely,
then what you're adding up is
only a finite amount of stuff.
The positive terms only add up
to a finite amount of stuff.
This is twice all the positive terms
and that only adds up to a finite number.
And likewise, the negative
terms only add up to a bunch
of finite amount of stuff.
So you take a finite
amount of positive stuff,
you subtract off a finite
amount of negative stuff,
and you get a total that makes sense.
So that isn't a rigorous proof.
That is.
This isn't.
But
it kinda explains what's going on.
So for example,
let's look at a sequence
One plus 1/4 minus 1/9 plus
1/16 plus 1/25 minus 1/36,
this looks like our one
over n squared series
except that some of
the terms are negative.
I've made every third term negative.
So you've got a positive,
a positive, a negative.
a positive, a positive, a negative,
a positive, a positive, a negative.
It's not an alternative series.
We don't know how to deal with
it as an alternating series.
All we know is that each term
is plus or minus one over n squared.
Well that means the absolute value of an
is one over n squared,
and the sum of the
absolute values converges.
That's because it's a p-series
with p equals two.
It converges by the Interval Test.
Well that means that our
original sum converged.
It doesn't tell us what
number this converges to,
but it does tell us this sum converges.
Another example was our harmonic series.
Our alternating harmonic series.
In this case, an was
plus or minus one over n.
The sum of the absolute values
was the sum of one over
n, and that diverged.
So this is an example of
a series that converges.
It converges because of the
Alternating Series Test.
But it does not converge absolutely.
We have a name for series that
converge but not absolutely.
They're said to converge conditionally.
We'll have a lot more to say
about series that converge conditionally
in the next video.
