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Finding the radius.

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[SOUND]

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Lets

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suppose that I've been given a power
series.

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Perhaps is the power series, the sum and
goes from one

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to infinity of x to the n divided by n
squared.

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How do I find the radius of convergence?

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Well lets try the ratio test.

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So I'm going to look for absolute
convergence.

10
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So I'm really trying to figure out for
which values

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of x that series converges.

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And I'm going to look at the ratio of the
n plus first term over the n term.

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So the n plus first term.

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It is x to the n plus 1 over n plus 1
squared.

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And I'm going to divide that by the n term
which is exactly what I've got there.

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X to the n over n squared.
Now I can simplify that fraction a bit.

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So this is.
The limit n goes to infinity of, I've

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got x to the n plus 1 over x to the n.
So I'll just write absolute value of x.

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And then I've got n plus 1 squared but
it's in the denominator of the numerator.

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And I've got n squared in the denominator
of the denominator.

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So I can write this as n squared over N
plus 1 squared.

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Now, what is this limit?

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Well when n is very large, this quantity
here is very close to one.

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And this x doesn't depend on n at all.

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So this limit is just the absolute value
of x.

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And this is the ratio between the n plus
first and the nth term.

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So to get absolute conversions of this
series,

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it's enough for ratio test that this be

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less than one.

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But I also know something about when the
series diverges.

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By the ratio test, when this limit which
is the absolute value of x.

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When that limit is bigger than one, then
this series diverges.

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Supporting it altogether what's the radius
of convergence.

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So to think about that let's draw a
diagram.

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You've got a number line.
And what I know is that when the

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absolute value of x is less than 1 then
the series converges absolutely.

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So that tells me that the series converges
when x is between minus 1 and 1.

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I also know that when the absolute value
of

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x is bigger than 1 that the series
diverges.

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That tells me the series diverges when x
is bigger than

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1 and the series diverges when x is less
than minus 1.

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So it converges in between here, it
diverges

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to right of this and diverges to the left
of this.

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Admittedly I haven't thought about what
happens at

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x equals minus 1 and x equals 1.

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But I don't need to if all care about is
knowing the radius of convergence.

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Alright, I'm thinking about this interval
being where the power series converges

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and maybe it convergences at minus 1,
maybe it converges at 1.

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But what's the radius of this interval or

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was this an interval centered at 0.
And its radius is 1.

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And that tells me that the

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radius of conversion is 1.

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[NOISE]