Why positive?
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Most of our convergence tests thus far have been assuming that the
terms in the series are positive or, or at least non-negative.
Yeah, when we were thinking about the comparison
test, the ratio test, even the root test.
Right in all of these, convergence tests, we're
trying to determine whether some series converges but
we're making this assumption that all of the terms are at least non-negative.
Why is this?
Let's think about why the ratio test in particular requires this
condition that the terms be non-negative when you're analyzing this series.
Remember what the ratio test tells us to do.
It tells us to look at this limit.
And say if this limit is less than 1, then the series converges.
And it shows that by doing
a comparison with this geometric series.
So indeed, I mean if, if you're given the ratio test and you look inside, what you
see is that the proof basically amounts to doing a comparison test for the geometric
series and the comparison test is really what requires this condition.
The terms be non-negative.
Okay, so most of these convergence tests are at some level of just reducing
the problem down to a comparison test. But that just raises another question.
Why does the comparison test require non-negativity of the terms?
Well, think back to how we proved the comparison test.
It's going to become clear that this condition, the
non-negativity of the a sub n's is extraordinarily important.
Remember what we did to prove the comparison test.
If you open up the comparison test,
it amounts to the monotone convergence theorem.
It's really just an application of the monotone conversions theorem.
Let's remember how we did this.
Right. How do we prove the comparison test?
So one direction went like this. I'm imagining I've got a series the sum of
the b sub n's converges, and the a sub n's are trapped between b sub n and 0.
Then I want to conclude that the sum of the a sub n's converges.
Well, because the a sub n's are all non-negative,
that tells me that the sequence of partial sums is non-decreasing.
Right.
This is where I'm using the crucial fact that the a sub n's are non-negative.
It's to get that the sequence of partial sums is non-decreasing.
And because b sub n is greater than or equal to a sub n, I also
know that the sequence of partial sums is
bounded by the value of this convergent series.
And consequently,
because the sequence of partial sums is monotone and bounded.
That tells me that the limit of the sequence of partial sums exists.
In other words, the sum of the a sub n's converges.
So non-negativity was important for the convergence
tests, because they relied on the comparison test.
And non-negativity was important for the comparison test, because the
comparison test is applying the monotone convergence theorem, and I
need non-negativity of the terms in the series in order
to know that the sequence of partial sums is monotone.
Now, that raises a question.
How can I analyze a series if the terms in the series aren't all non-negative?
And indeed, that very question is going to be a major theme for what is to come.
What are we supposed to do with series when
some of the terms are positive and some of the
terms are negative?
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