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Why positive?

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[SOUND]

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Most of our convergence tests thus far
have been assuming that the

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terms in the series are positive or, or at
least non-negative.

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Yeah, when we were thinking about the
comparison

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test, the ratio test, even the root test.

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Right in all of these, convergence tests,
we're

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trying to determine whether some series
converges but

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we're making this assumption that all of
the terms are at least non-negative.

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Why is this?

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Let's think about why the ratio test in
particular requires this

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condition that the terms be non-negative
when you're analyzing this series.

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Remember what the ratio test tells us to
do.

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It tells us to look at this limit.

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And say if this limit is less than 1, then
the series converges.

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And it shows that by doing

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a comparison with this geometric series.

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So indeed, I mean if, if you're given the
ratio test and you look inside, what you

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see is that the proof basically amounts to
doing a comparison test for the geometric

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series and the comparison test is really
what requires this condition.

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The terms be non-negative.

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Okay, so most of these convergence tests
are at some level of just reducing

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the problem down to a comparison test.
But that just raises another question.

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Why does the comparison test require
non-negativity of the terms?

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Well, think back to how we proved the
comparison test.

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It's going to become clear that this
condition, the

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non-negativity of the a sub n's is
extraordinarily important.

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Remember what we did to prove the
comparison test.

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If you open up the comparison test,

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it amounts to the monotone convergence
theorem.

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It's really just an application of the
monotone conversions theorem.

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Let's remember how we did this.

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Right.
How do we prove the comparison test?

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So one direction went like this.
I'm imagining I've got a series the sum of

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the b sub n's converges, and the a sub n's
are trapped between b sub n and 0.

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Then I want to conclude that the sum of
the a sub n's converges.

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Well, because the a sub n's are all
non-negative,

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that tells me that the sequence of partial
sums is non-decreasing.

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Right.

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This is where I'm using the crucial fact
that the a sub n's are non-negative.

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It's to get that the sequence of partial
sums is non-decreasing.

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And because b sub n is greater than or
equal to a sub n, I also

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know that the sequence of partial sums is

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bounded by the value of this convergent
series.

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And consequently,

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because the sequence of partial sums is
monotone and bounded.

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That tells me that the limit of the
sequence of partial sums exists.

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In other words, the sum of the a sub n's
converges.

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So non-negativity was important for the
convergence

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tests, because they relied on the
comparison test.

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And non-negativity was important for the
comparison test, because the

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comparison test is applying the monotone
convergence theorem, and I

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need non-negativity of the terms in the
series in order

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to know that the sequence of partial sums
is monotone.

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Now, that raises a question.

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How can I analyze a series if the terms in
the series aren't all non-negative?

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And indeed, that very question is going to
be a major theme for what is to come.

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What are we supposed to do with series
when

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some of the terms are positive and some of
the

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terms are negative?

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