Monotonicity matters.
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Do you want
to meet a statement of the alternating series test?
So here is what the alternating series test tells us.
If I've got a sequence, the terms of which are decreasing, and they're all positive.
And the limit of a sub n as n approaches infinity is zero,
then the alternating series that I build using this sequence a sub n.
The sum n goes from one to infinity of negative one to the n plus one,
times a sub n. So this is an alternating series.
This alternating series converges.
I've been assuming that the sequence a sub n was decreasing.
And I used that to get that the even and the odd partial sums were monotone.
But what happens if I get rid of that condition.
What happens if I get rid of the condition that a sub n is decreasing?
Well, let's break the alternating series test by removing
this condition.
I'm going to remove the condition that the sequence a sub n be decreasing.
I'll call this the Broken Alternating Series Test.
I'm just going to assume that the a sub
n's are positive and that their limit is zero.
And if I just make these two assumptions, is
it then the case that this alternating series necessarily converges?
So to see how broken this really is,
the question is this: can you think of an alternating series
where the terms are going to zero, but the series diverges?
Well, here's an example.
It's the sequence a sub n.
And the definition depends on whether n is odd or even.
So if n is odd, it's one over n plus one divided by two.
And if n is even, it's one over n over two squared.
Well, what does this look like? I'm going to use this sequence
to build an alternating series.
And what is this alternating series, then then, looking like?
Well, I plug in n equals one, and that's odd, so
it's one over one plus one over two plus one over one.
I plug in n equals two. I get one over two over two squared.
But that's going to come with a minus sign.
I plug in n equals three.
It'll be three is odd, so it's one over three plus one over two, it's a half.
I mean, here's some of the terms. I just started
writing them down.
Alright, so it's one over one, and then minus one over one
squared, plus one over two minus one over two squared, plus one
over three minus one over three squared, plus one over, I mean,
what this really amounts to is sort of weaving together two different series.
Alright, the positive terms are the harmonic series, and the
negative terms in this alternating series are the p series
where p equals two. And the limit of the nth term is zero.
Well, specifically, the limit of the odd terms here, it's
just the limit of one over n, and that is zero.
And the limit of the even terms here, it's just
the limit of one over n squared, and that's also zero.
So putting together these two facts just the limit of a sub n is equal to zero.
But the sequence,
it's not decreasing.
If I look at the terms, all right, just
the first few terms of this sequence a sub n.
I mean, these are not decreasing. I mean, look.
One half and then one fourth, but then one third right?
It's going up here, down here, up here, down here, up here.
It's not a decreasing sequence. And the alternating series diverges.
So I want to look at the sequence of partial sums s and two n.
The sum k goes from one to two n minus one to the k plus one times a sub k.
Now, the whole thing about this series is that it's weaving
together the harmonic series and a p series with p equals two.
So, this partial sum can be written like this.
The odd terms are the harmonic series.
And the even terms are giving me this p series, with p equals two.
The odd terms are positive. And the even terms are negative.
So this partial sum is this.
It's the sum k goes from one to n of one over k
minus the sum k goes from one to n of one over k-squared.
Now, the whole point here is that the harmonic
series diverges and this is a convergent p series.
So that means by taking n large enough, I can make this harmonic series, or this
partial sum for the harmonic series, as large as I'd like.
But no matter how big I take n, this convergent p-series never
exceeds the true value of this p-series, which is pi squared over six.
So what happens then, when n is really large, right?
This term becomes very, very large, as large as I'd like.
This term never exceeds the value of the convergent p-series where p equals two.
And that means
that the limit of these even partial sums is infinite.
And consequently, this series diverges.
because the limit of the partial sums doesn't converge.
This is a particular philosophy, by which you can study the whole world.
If you wouldn't know that car engines are important.
When you take the car engine out, then the car doesn't go.
So it must have been important.
Well this same sort of game is being played with the alternating
series test.
I took out a part of the alternating series test.
I've removed the assumption that a sub n is decreasing.
And I saw that the alternating series test didn't work anymore.
So although I don't emphasize it all the time, the fact that the a sub
n sequence is a decreasing sequence, really is
important when you're applying the alternating series test.
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