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Conditional convergence.

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Not every convergence series converges
absolutely.

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Seeing that absolute convergence and just
plain old convergence

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are related, right, absolute convergence
implies plain old convergence.

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But it turns out that this doesn't go the
other way.

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It's not the case that convergence implies
absolute convergence.

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So we'll give a name to this situation.

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So here's the definition, a series is
conditionally convergent

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if the series converges, but the sum of
the absolute values diverges.

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In other words, conditional convergence
means

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the series converges but not absolutely.

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Let me draw a diagram of the situation.

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So if I start out considering all series,
once I start thinking

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about convergence, right, that separates
series

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into two different kinds of series, right?

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The divergent series and

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the convergent series.

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But now I've got this more refined notion
of convergence, absolute convergence.

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So I can subdivide the convergent series
into two kinds of series,

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those that are absolutely convergent, and

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those that are just conditionally
convergent.

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I mean, they still converge, but they
don't converge absolutely.

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Now, can I think of anything in

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the conditionally convergent part of that
diagram?

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Can I think of any conditionally

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convergent series at all?
Well, here's an example.

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The sum n goes from 1 to infinity of
negative 1 to the nth power divided by n.

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I know that series is not absolutely
convergent.

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Well I know this series is not absolutely
convergent, for the following reason.

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Alright?

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I can look at the sum, n goes from 1 to
infinity

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of the absolute value of negative 1 to the
n over n.

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Well, what is that?

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That's just the sum n goes from 1 to
infinity.

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What's the absolute value of minus 1 of,
to the n?

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That's just 1, this is just 1 over n, this

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is just the harmonic series, and the
harmonic series diverges.

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And since the sum of the absolute values
diverges, it's exactly

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what it means to say that the series does
not converge absolutely.

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But the series does converge.

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Yeah, does converge and

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in fact the sum n goes from 1 to infinity
of minus 1 to the

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n over n ends up converging to negative
the natural log of 2.

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And since this series does converge, but
it doesn't converge absolutely.

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This is an example of a conditionally
convergent series.

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We don't quite yet have the tools to show
that, but we're awfully close.

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