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Alternating series are awesome.

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[SOUND].

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Why do we care about alternating series?

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Well, if you're trying to analyze the
convergence of a

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series and all of the terms in the series
are non-negative,

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then you can break out all your usual
convergence tests, the

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comparison test, the root test, the ratio
test, the integral test.

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What about series where not all the terms
are non-negative?

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So maybe I'm trying to analyze a series
where this doesn't happen.

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It's not

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the case that all the terms are
non-negative, but I've

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got some positive terms and some negative
terms in my series.

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Well then what am I supposed to do?

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Well one thing I could do in this case is
instead

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of analyzing this series directly I could
look at this series.

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The sum n goes from one to infinity of the
absolute values of a's of n.

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Try to show that this series converges
absolutely.

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And therefore,

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this series would converge.

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And what about series that don't converge
absolutely?

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Let's suppose I analyze this series, the
sum, little n goes from

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one to infinity, negative one to the n
plus one, divided by n.

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Now my first inclination would be to take
a look at this series, the sum of the

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absolute values of these terms, with the
hopes that

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I maybe could prove that this thing
converges absolutely.

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But that's not true.

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This series doesn't converge absolutely,
because this series diverges.

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What is the absolute value of negative one
to the n plus one over n?

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This.
It's just 1 over n.

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This the harmonic series, the harmonic
series

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diverges so this series does not converge
absolutely.

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So we've gotta do something else.

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And indeed, the ultimate in series tests
comes to save the day.

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I'll rewrite this series like this, as the
sum

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n goes to infinity of negative 1 to the n
plus

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1 times a sub n Where a-sub-n is 1 over n.

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And now what I note is that the sequence
a-sub-n is a decreasing sequence

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all of the terms of that sequence are
positive, and the limit of a-sub-n is 0.

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Just the limit of 1 over n as it
approaches infinity is 0 And

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that means, by the alternating series
test,

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the series that I'm studying here
converges.

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Now I've just shown that it doesn't
converge absolutely,

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so what the opening series test is
actually showing.

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Is it this series converges conditionally.

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This is partly why alternating series are
so important.

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Because of the alternating series test, we
can

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prove that an alternating series converges
without using our other conversions tests

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on the series of the absolute values,
without proving absolute conversions.

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We don't honestly have that many other
tools for showing that a series, some

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of whose terms are positive, some of whose
terms are negative, converges at all.

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Normally the way we'd approach those is by
showing that they converge absolutely.

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So what are we supposed

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to do about those series which don't
converge absolutely but do converge?

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What can we do about the conditionally
convergent series in our world?

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Well, the alternating series test is a
great tool in our toolbox.

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And the alternating series test gives us
more than just convergence.

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Suppose I want to approximate the value of
this series.

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What could I do?

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It's an alternating series, so I know that
the even partial

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sums in this case will be underestimates
of

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the odd partial sums that will be
overestimates.

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So, here is one of those odd partial sums,
the sum of the first 3 terms.

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1 over 1, minus 1 over 2, plus 1 over 3.

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That's an overestimate, the true value of
this series.

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And here's an even partial sum, sum of the
first four terms.

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And that's an underestimate, the true
value of this series.

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And then I could actually figure out what
these are, right?

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1 minus a half is a half plus a third.
that's 5 6ths.

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And here I've got 5 6ths minus a quarter.
That's 7 12ths.

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So I know that 7 12ths is less than or
equal to the

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true value of the series, is less than or
equal to 5 6ths.

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Let me rewrite five sixths.

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Alright I could call five sixths, ten
twelfths.

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It makes it a little bit clearer.
I think that it's

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[LAUGH]

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actually bigger than seven twelfths.
Now it turns out that this series is value

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is actually log 2.
So what I've done here is shown that log

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2.
Is between 7 12ths and 10 12ths, right?

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What I've really done is I've shown this.
Alright.

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I've got this inequality now but this is
just coming

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because I've expressed log 2 as the value
of an alternating

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series, and alternating series provide
these

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very convenient error bounds on my
estimates.

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I could multiply everything here by 12 and
that would tell

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me that 7 is less than or equal to 12
times

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log 2 which by properties of log rhythms
is log of

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2 to the 12th and that's less than or
equal to 10.

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And then I

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could e to the all three of these things,
right.

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I could apply the exponential function to
all three of these things,

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and I'd find out that e to the 7th is less
than

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or equal to 2 to the 12th as e to the log

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undoes the log, is less than or equal to e
to the 10th.

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And 2 to the 12th is 4096, so what I've
shown just by playing

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around with alternating series is this.
That 4096

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is between these two numbers, and I mean
this isn't great, I mean these bounds

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aren't fantastic, I mean I didn't add up
very many terms in this series, right?

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But still, I think it demonstrates the
principle that one of the coolest

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things about alternating series is that
alternating series provide these

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convenient bounds for you.

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[NOISE]