Alternating series are awesome.
[SOUND].
Why do we care about alternating series?
Well, if you're trying to analyze the convergence of a
series and all of the terms in the series are non-negative,
then you can break out all your usual convergence tests, the
comparison test, the root test, the ratio test, the integral test.
What about series where not all the terms are non-negative?
So maybe I'm trying to analyze a series where this doesn't happen.
It's not
the case that all the terms are non-negative, but I've
got some positive terms and some negative terms in my series.
Well then what am I supposed to do?
Well one thing I could do in this case is instead
of analyzing this series directly I could look at this series.
The sum n goes from one to infinity of the absolute values of a's of n.
Try to show that this series converges absolutely.
And therefore,
this series would converge.
And what about series that don't converge absolutely?
Let's suppose I analyze this series, the sum, little n goes from
one to infinity, negative one to the n plus one, divided by n.
Now my first inclination would be to take a look at this series, the sum of the
absolute values of these terms, with the hopes that
I maybe could prove that this thing converges absolutely.
But that's not true.
This series doesn't converge absolutely, because this series diverges.
What is the absolute value of negative one to the n plus one over n?
This. It's just 1 over n.
This the harmonic series, the harmonic series
diverges so this series does not converge absolutely.
So we've gotta do something else.
And indeed, the ultimate in series tests comes to save the day.
I'll rewrite this series like this, as the sum
n goes to infinity of negative 1 to the n plus
1 times a sub n Where a-sub-n is 1 over n.
And now what I note is that the sequence a-sub-n is a decreasing sequence
all of the terms of that sequence are positive, and the limit of a-sub-n is 0.
Just the limit of 1 over n as it approaches infinity is 0 And
that means, by the alternating series test,
the series that I'm studying here converges.
Now I've just shown that it doesn't converge absolutely,
so what the opening series test is actually showing.
Is it this series converges conditionally.
This is partly why alternating series are so important.
Because of the alternating series test, we can
prove that an alternating series converges without using our other conversions tests
on the series of the absolute values, without proving absolute conversions.
We don't honestly have that many other tools for showing that a series, some
of whose terms are positive, some of whose terms are negative, converges at all.
Normally the way we'd approach those is by showing that they converge absolutely.
So what are we supposed
to do about those series which don't converge absolutely but do converge?
What can we do about the conditionally convergent series in our world?
Well, the alternating series test is a great tool in our toolbox.
And the alternating series test gives us more than just convergence.
Suppose I want to approximate the value of this series.
What could I do?
It's an alternating series, so I know that the even partial
sums in this case will be underestimates of
the odd partial sums that will be overestimates.
So, here is one of those odd partial sums, the sum of the first 3 terms.
1 over 1, minus 1 over 2, plus 1 over 3.
That's an overestimate, the true value of this series.
And here's an even partial sum, sum of the first four terms.
And that's an underestimate, the true value of this series.
And then I could actually figure out what these are, right?
1 minus a half is a half plus a third. that's 5 6ths.
And here I've got 5 6ths minus a quarter. That's 7 12ths.
So I know that 7 12ths is less than or equal to the
true value of the series, is less than or equal to 5 6ths.
Let me rewrite five sixths.
Alright I could call five sixths, ten twelfths.
It makes it a little bit clearer. I think that it's
[LAUGH]
actually bigger than seven twelfths. Now it turns out that this series is value
is actually log 2. So what I've done here is shown that log
2. Is between 7 12ths and 10 12ths, right?
What I've really done is I've shown this. Alright.
I've got this inequality now but this is just coming
because I've expressed log 2 as the value of an alternating
series, and alternating series provide these
very convenient error bounds on my estimates.
I could multiply everything here by 12 and that would tell
me that 7 is less than or equal to 12 times
log 2 which by properties of log rhythms is log of
2 to the 12th and that's less than or equal to 10.
And then I
could e to the all three of these things, right.
I could apply the exponential function to all three of these things,
and I'd find out that e to the 7th is less than
or equal to 2 to the 12th as e to the log
undoes the log, is less than or equal to e to the 10th.
And 2 to the 12th is 4096, so what I've shown just by playing
around with alternating series is this. That 4096
is between these two numbers, and I mean this isn't great, I mean these bounds
aren't fantastic, I mean I didn't add up very many terms in this series, right?
But still, I think it demonstrates the principle that one of the coolest
things about alternating series is that alternating series provide these
convenient bounds for you.
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