Conditional convergence.
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Not every convergence series converges absolutely.
Seeing that absolute convergence and just plain old convergence
are related, right, absolute convergence implies plain old convergence.
But it turns out that this doesn't go the other way.
It's not the case that convergence implies absolute convergence.
So we'll give a name to this situation.
So here's the definition, a series is conditionally convergent
if the series converges, but the sum of the absolute values diverges.
In other words, conditional convergence means
the series converges but not absolutely.
Let me draw a diagram of the situation.
So if I start out considering all series, once I start thinking
about convergence, right, that separates series
into two different kinds of series, right?
The divergent series and
the convergent series.
But now I've got this more refined notion of convergence, absolute convergence.
So I can subdivide the convergent series into two kinds of series,
those that are absolutely convergent, and
those that are just conditionally convergent.
I mean, they still converge, but they don't converge absolutely.
Now, can I think of anything in
the conditionally convergent part of that diagram?
Can I think of any conditionally
convergent series at all? Well, here's an example.
The sum n goes from 1 to infinity of negative 1 to the nth power divided by n.
I know that series is not absolutely convergent.
Well I know this series is not absolutely convergent, for the following reason.
Alright?
I can look at the sum, n goes from 1 to infinity
of the absolute value of negative 1 to the n over n.
Well, what is that?
That's just the sum n goes from 1 to infinity.
What's the absolute value of minus 1 of, to the n?
That's just 1, this is just 1 over n, this
is just the harmonic series, and the harmonic series diverges.
And since the sum of the absolute values diverges, it's exactly
what it means to say that the series does not converge absolutely.
But the series does converge.
Yeah, does converge and
in fact the sum n goes from 1 to infinity of minus 1 to the
n over n ends up converging to negative the natural log of 2.
And since this series does converge, but it doesn't converge absolutely.
This is an example of a conditionally convergent series.
We don't quite yet have the tools to show that, but we're awfully close.
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