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Half open intervals.

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Well, as an example, let's consider this
series, the sum, n

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goes from 1 to infinity, of x to the n,
over n.

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Well it's a power series, so what's its
interval of convergence?

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Well fore we address that question, we ask
the easier question.

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What's its radius of convergence?
Well, apply the ratio test.

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So, I'm going to think about where this
series converges

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absolutely, for which values of x does it
converge absolutely.

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So I'm wondering when does this converge,
and the

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ratio test tells me to look at this limit.

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The limit, as n approaches infinity of the
n plus first term, which is x to

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the n plus 1 over n plus 1, divided by the
nth term which is x to the n over n.

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And we're looking at this with absolute

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value bars and asking about absolute
convergence.

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So the ratio test asks me to consider for

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which values of x is this limit less than
1?

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Well I can simplify this limit somewhat.

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All right, this is the limit as n
approaches infinity.

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I've got x to the n plus one over x to the
n here.

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Those cancel and just leave me with an x.

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And here I've got n plus 1 in the
denominator of

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the numerator and an n in the denominator
of the denominator,

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this ends up being n over n plus 1.
So now, what's this limit?

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Well, the limit of n over n plus 1, as n
approaches infinity, that's 1 and

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this is just a constant, so this limit of
the constant as far as n is concerned.

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So this limit is just the absolute value
of x.

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So this series converges absolutely when
the absolute value of x is less

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than 1, and it diverges when the absolute
value of x is bigger

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than 1.
So what is the radius of convergence?

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Well, this is telling me that the radius
of convergence is 1, so

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I could plot the points on the number line
where this series converges.

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And what I know thus far, is that it
converges, when the absolute value of x

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is less than 1, meaning that x is between
minus 1 and 1, and it diverges when

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the absolute value of x is bigger than 1.

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So I know it diverges out here, and it
diverges down here.

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What about the end points?

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Well, exactly.

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Alright, what happens when x is equal to
1, right?

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Does the series converge or diverge at 1?
What happens when x is equal to minus 1?

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Does the series converge or diverge at
minus 1?

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All I know so far is

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that it converges absolutely between minus
1 and 1 and it diverges out here, but

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I haven't actually addressed the question
of whether

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or not this series converges at the
endpoints.

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We can plug in x equals 1 and then
recognize the series.

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Yeah, so, if we're looking at this series,
the sum n goes from 1

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to infinity of x to the n over n, and I
plug in x

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equals 1, what do I get?

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Well then, this is just the sum n goes
from 1

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to infinity of 1 to the n, which is 1 over
n.

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Does this series converge or diverge?

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That's the harmonic series, and the
harmonic series

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diverges, which means that this series,
when x is equal to one, diverges.

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Now, what about x

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equals negative one?

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Well, in that case, I'm going to plug in
minus 1 here, and I'll get the sum n goes

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from 1 to infinity of minus 1 to the n
over n, and

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that's the alternating harmonic series,
and that converges, albeit conditionally.

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We can summarize this.

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So putting this all together, we can write
the following.

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We can say

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that the interval of convergence is the
half open interval

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minus 1 to 1 but closed on the minus 1
side because this series

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converges here but doesn't converge at 1.

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So the interval of convergence is this
half open interval.

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And note

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just how complicated this was.

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We've got our interval of convergence, and
in

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the interior of that interval, we've got
absolute convergence.

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And it wasn't too hard to figure out the
radius of that

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interval, but the story became way more
complicated at the end points.

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We had one endpoint where the series
converged and another endpoint where

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the series diverged, and in general, this
is how it's going to work out.

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It won't be hard for you to find the
radius of convergence,

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but it might be really painful to analyze
the story at the endpoints.

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And it's possible that the series could
diverge at both endpoints.

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It might converge at both endpoints.

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It might just converge at one endpoint.

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Story at the endpoints is more
complicated.

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