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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