### Video Transcript

Consider the power series the sum
from π equals zero to β of π times two π₯ to the πth power. Determine the radius of convergence
of the power series and determine the interval of convergence of the power
series.

To test for convergence of this
power series, we could use the ratio or the root test. Letβs look at the root test. The part of the test weβre
interested in says suppose we have a series the sum of π π. If the limit as π approaches β of
the πth root of the absolute value of π π, which can of course be alternatively
written as the absolute value of π π to the power of one over π, is less than
one. Then the series is absolutely
convergent and hence convergent.

So we define π π for our series
to be π times two π₯ to the πth power. Weβre looking to establish where
the limit as π approaches β of the absolute value of π times two π₯ to the πth
power to the power of one over π is less than one. We know that finding an πth root
doesnβt change the sign. So we can rewrite this and say that
the limit as π approaches β of the absolute value of π to the power of one over π
times two π₯ to the πth power to the power of one over π must be less than
one. Two π₯ to the πth power to the
power of one over π is simply two π₯. And so our limit is as shown.

We know that two π₯ is independent
of π. So we can take the absolute value
of two π₯ outside of our limit. The problem we have now is that if
we try to evaluate the limit as π approaches β of the absolute value of π to the
power of one over π. We get the absolute value of β to
the power of one over β, which is β to the power of zero. And we know thatβs indeterminate
form. But we might recall that we can
write π to the power of one over π as π to the power of the natural log of π
over π. And we know that as π approaches
β, the natural logarithm of π over π approaches zero. And so the limit as π approaches β
of the absolute value of π to the power of one over π is π to the power of zero,
which is simply one.

Of course we donβt need the
absolute value signs for one. And so we see weβre interested in
when the absolute value of two π₯ is less than one. Since two is a purely positive
number, we can divide through. And we see that the absolute value
of π₯ is less than one-half. These are the values of π₯ for
which the series converges. And we can therefore say that the
radius convergence of our power series is one-half.

Another way of representing the
absolute value of π₯ being less than one-half is to say that π₯ must be greater than
negative one-half and less than one-half. And so weβve determined an interval
for convergence. But we do need to check if the
power series converges or diverges at the end points of our interval. In other words, when π₯ is equal to
negative one-half or π₯ is equal to one-half.

Weβre going to plug these into our
original power series and see if those series converge or diverge using alternative
tests. Letβs clear some space. Weβll begin by letting π₯ equal
negative one-half. The series is the sum from π
equals zero to β of π times two minus negative one-half to the πth power. And this becomes π times negative
one to the power of π.

To establish whether this series
converges or diverges, we could use the alternating series test with π π equals
π. Since the limit as π approaches β
of π π is not equal to zero, we can say that this series ultimately diverges. So letβs try when π₯ is equal to
one-half. We have the sum from π equals zero
to β of π times two times a half to the πth power. Thatβs π times one to the πth
power. And of course one to the πth power
will always be one. So we have the sum from π equals
zero to β of π.

This time, we can actually apply
the πth term divergence test. The limit as π approaches β of π
is β. Thatβs not equal to zero. And so, once again, when π₯ is
equal to a half, our series diverges. And so the interval of convergence
of our power series is the open interval negative one-half to one-half.

In this video, we saw that a series
of the form the sum of π π times π₯ minus π to the πth power for values of π
between zero and β is called a power series about π. We also saw that we can use the
ratio test and root test to find the radius of convergence and interval of
convergence of our power series. But that itβs also important that
we should test the end points of the interval for convergence or divergence.