Absolute convergence.
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What do I even mean by absolute convergence?
Well here's the definition.
Right, so defintion,
the series, I'll write down a series, the sum n goes from 1 to infinitiy of
a sub n Converges absolutely.
So this is the term that I'm defining. Defining absolute conversion.
So I'm going to say the series converges absolutely if what happens?
If the series,
just the sum n goes from 1 to infinity of the absolute values of the a sub n's just
plain old converges in the usual sense. What's an example of this?
While ago, so that this series.
The sum n goes from 1 to infinity of the absolute
value of sine n over 2 to the n; that series converges.
you could prove that by doing a comparison test with the geometric
series 1 over 2 to the n.
Alright, so that series converges, and that means that this
series, the sum n goes from 1 to infinity of
just sine n over 2 to the n, no more
absolute value now, just sine n over 2 to the n.
That series converges absolutely, or we might say is absolutely convergent, just
because this series, the sum of
the absolute value of this quantity, converges.
But with a name
like absolute convergence, you would hope.
That an absolutely convergent series is also
just convergent in the regular sense of converging.
So that's a theorem.
If the sum of the absolute values of the a sub
n's converges, then just the sum of the a sub n's converges.
In other words, if a series converges absolutely then just plain old converges,
let's prove that. Okay yes so how's this proof going to go.
Well the proof is going to start out
with my assuming that the series converges absolutely.
Meaning I'm going to assume that the sum of
the absolute value of the a sub n's converges.
And then somehow you know, I want to do something.
I don't know what yet, so I've got some cloud with question marks.
And then, I eventually want to conclude that the original series converges.
So what goes in these question marks, right?How
do I get from this assumption to this conclusion.
Well the first thing I can note is that if this
series converges Then the sum of twice those terms also converges.
Why is that important?
Well here is the key fact, this is sort of a trick if you like.
But what it's saying is that 0 is less than or equal to a
sub n plus the absolute value of a sub n is less than or
equal to twice the absolute value of a sub n.
Why is this even true?
Well, think about the possibilities. Maybe a sub n is negative.
And if a sub n is negative, then I've got a
negative number plus well, the absolute value of that negative number.
These two things cancel, and I just get zero.
And then certainly this is true.
Zero is less or equal to zero, is
less than or equal to twice some positive number.
Well what if a sub n were positive?
Well, then I'd have 0 is less than or
equal to a positive number plus itself, cause the absolute
value of a positive number is just that same
number, is less than equal to twice that same number.
Well, in that case, these are equal right, I've got a positive number plus itself.
That's the same as twice that positive number.
So if a sub n's positive, this is true if a sub
n's negative, this is true if a sub n's 0, this is
just 0 less than or equal to 0 plus 0 less than equal to 2 times 0.
So whether a sub n is positive, whether it's 0, whether
it's negative, no matter what happens, this is a true statement.
Now we can apply the comparison test.
So by the comparison test this series converges right.
The sum of these converges the side angles from 1 to infinity
of a sub n plus the absolute value of a sub n.
Well that's term wise less than this convergent series and all
the terms are non negative so that means this series converges.
but now our original series can be written in terms of this
series and the sum of the absolute values in the a sub n's.
So what I got is this series converges, this
series converges, but the difference of these two series is
the series that I'm originally interested in, right.
And the difference of convergent series converges so that mean that
the series just the sum of the a sub n's converges.
Which is exactly what I wanted to show.
So we prove the theorem. Alright, if a series converges absolutely,
then it just plain old converges. And that really justifies the terminology.
I'm calling this, well everybody calls this, absolute
convergence, because it's a strong kind of convergence.
Absolute convergence implies convergence, so absolute convergence
is even stronger than just regular old convergence.
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