We want to find the interval
on rays of convergence for
the power series k equals 0
to infinity of x to the k.
So we're gonna test for
absolute convergence using the ratio test.
So we're going to find the limit as
k goes to infinity of the absolute
value of x to the k + 1 divided by
the absolute value of x to the k.
So this equals the limit as k goes to
infinity of the absolute value of x.
And, this absolute value of x, there's no
k there, so we can pull it out in front so
that we get the absolute value of x times
the limit as k goes to infinity of 1,
and the limit as k goes to
infinity of 1 is just 1.
So we get the absolute value of x times 1.
[SOUND] So,
this tells us that the limit as k goes
to infinity of the absolute
value of a sub k + 1
divided by the absolute value
of a sub k is less than
1 when the absolute value
of x is less than 1.
So therefore, we converge for
x between -1 and 1.
And we can write this in
interval notation as -1 to 1.
So the series Converges.
For x in -1 to 1.
So this means that our
radius of convergence,
since we're starting at 0 is the absolute
value of 1- 0, which equals 1.
So drawing this on a number line.
We see that our radius of convergence.
Here's our radius.
So our radius, R is 1.
So now we need to know if we
converge at the endpoints.
So, do we converge at, x = -1 or x = 1?
So to test x = 1 we're gonna
plug that into our power series.
So we get the series k = 0 to infinity
of 1 to the k and this diverges.
I'm gonna let you think
about why this diverges.
X = -1,
we have the series k = 0 to infinity
of -1 to the k, which also diverges.
And again, you think about that,
why that is on your own.
So that tells us that our interval
of convergence is -1 to 1,
not including the endpoints.
We want to find the interval and
radius of convergence of the power series
k = 0 to infinity of x + 3 to the k.
So here our center.
Is -3 and
we get that because x- -3 to the k.
Our power series are always in the form
x- a where a is our center, so
here our a is -3.
So using the ratio test for
absolute convergence we
have the limit as k goes to
infinity of the absolute
value of x + 3 to the k
+ 1 divided by the absolute
value of x + 3 to the k.
And this equals the limit as k
goes to infinity of the absolute
value of x + 3 which equals
the absolute value of x + 3.
So, this tells us that we converge when
the absolute value of x
+ 3 is less than 1 or
when x + 3 is between -1 and 1.
And solving this inequality we
have that x is between -4 and -2.
So our radius of convergence.
R equals the absolute value of
-2- (-3) which gives us 1.
So our radius of convergence is 1.
We need to test our endpoints.
So we're testing x = -4 and x = -2.
Looking at X =-
4 we get the series k = 0 to
infinity of -4 + 3 to the k.
Which is the series k = 0 to infinity
of -1 to the k which diverges.
And looking at x = -2,
we get the series k =
0 to infinity of -2 + 3 to the k.
Which is the series k = 0
to infinity of 1 to the k,
which also diverges, so
that means that our interval
of convergence is -4 to -2 where
our endpoints are not included.
So on a number line,
we were centered at -3,
our radius was 1, so
we went out to -4 to 2,
and we are not including our endpoints.
We want to find the interval and
radius convergence for
the series k = 0 to
infinity of 3x- 4 to the k.
So first we wanna find our center.
So we know for a power series
we're in the form x- a to the k.
So to find our center we need
to get ourselves into this form.
So first we're going to take out 3.
So that we have 3 times a quantity
x- minus 4 divided by 3.
And that tells us that our
center is four-thirds.
So we're going to use the ratio test
to test for absolute convergence.
So we have the limit as
k goes to infinity of
the absolute value of 3x- 4 to the k + 1
divided by the absolute
value of 3x- 4 to the k,
and that gives us the limit as k goes to
infinity of the absolute value of 3x- 4.
And that just equals
the absolute value of 3x- 4.
So we converge.
When the absolute value
of 3x- 4 is less than 1.
So that means that 3x- 4 is between -1 and
1.
So solving this inequality we get 3 is
less than 3x, which is less than 5.
So 1 is less than x,
which is less than five-thirds.
And define our radius of convergence.
R equals absolute value of
five-thirds minus our center,
which was four-thirds, which is one-third.
So our radius of convergence is one-third,
and now I need to test our endpoints.
So we're testing x = five-thirds and 1.
So testing x = five-thirds,
we get the series k = 0 to infinity of
3 times five-thirds- 4 to the k,
which equals the series k = 0 to infinity
of 1 to the k, which diverges.
And for k = 1,
we get the series k = 0 to infinity
of 3 x 1- 4 to the k which
is a series k = 0 to
infinity of -1 to the k,
which also diverges.
So our interval of convergence
is interval one to five-thirds,
where the endpoints are not included.
So on our number line,
we have four-thirds was our center,
up to five-thirds, going down to one.
And our endpoints were not included.
We want to find the interval and
radius of convergence of the power
series k = 0 to infinity
of x to the k divided by k.
So on this one, our center is 0.
So we're gonna use the ratio test to
test for absolute convergence and
we get the limit as k goes
to infinity of the absolute
value of x to the k + 1
divided by k + 1 divided
by the absolute value of
x to the k divided by k.
And we can rewrite this as
the limit as k goes to infinity
of the absolute value of x
times k divided by k + 1.
So pulling x, the absolute value
of x out in front of the limit,
we get absolute value of x
times the limit as k goes
to infinity of k divided by k + 1 and
this limit is 1.
So we get the absolute value of x.
So this series converges when
the absolute value of x is less than 1.
So that means the x is between -1 and 1.
So our radius of convergence
R = 1- 0 which is 1.
So now we need to test our endpoints.
So we're going to be testing x = -1 and
x = 1.
For x = 1.
We get the series k = 1 to infinity of 1
to the k divided by k, so the series k = 1
to infinity of 1 over k and this diverges.
If we look at x = -1, we get the series
k = 1 to infinity of -1
to the k divided by k.
And this will converge by
the alternating series test.
We'll leave you to figure out the details.
So this tells us that our
interval of convergence
Is going to be -1 to 1
where -1 is included, so
on our number line,
our center was 0, go up to 1,
down to -1 and
-1 is included but 1 is not.
Here we want to find the interval and
radius of convergence of k = 0 to
infinity of x to the k times k to the k.
So here our center is 0.
And we're going to use
the root test on this one.
To test for absolute conversions we
find the limit as k goes to infinity
of the kth root of the absolute value
of x to the k, times k to the k.
And this equals the limit as k
goes to infinity of the kth root
of the absolute value of x to the k
times the kth root of k to the k.
So then we're going to end up with
the limit as k goes to infinity
of the absolute value of x times k, so
we can pull the absolute value
of x out in front of the limit.
So we have the absolute value of x times
the limit as k goes to infinity of k.
And the limit as k goes to
infinity of k equals infinity.
So we have infinity is
always bigger than 1 for
all x except at the center.
X = 0.
Because when x = 0,
we would have the series
k = 0 to infinity of k to
the k times 0 to the k,
which is series k = 0 to infinity of 0,
which = 0.
So this tells us that our radius,
R Equals 0, and
our interval of convergence
is just the single point 0.
So on our number line it's
just the point 0, our center.
We will define the interval and radius
of convergence of the power series k
= 0 to infinity of -1 to the k times
x to the k divided by 2k factorial.
So here our center is 0.
And we're going to use the ratio test
to test for absolute convergence.
So we find the limit as
k goes to infinity of
the absolute value of
-1 to the k + 1 times x
to the k + 1 divided by
2 times k + 1 factorial
divided by the absolute
value of -1 to the k,
times x to the k, divided by 2k factorial.
So we want to rewrite
this as a limit as k goes
to infinity of the absolute
value of -1 to the k +
1 divided by the absolute
value of -1 to the k,
times the absolute value of x to the k + 1
divided by the absolute
value of x to the k,
times 2k factorial divided by 2
times the quantity k + 1 factorial.
So first thing to note, that this
first part, the absolute value of -1
to the k + 1 divided by the absolute value
of -1 to the k, those are absolute values.
This is always going to be 1.
The second part, the absolute
value of x to the k + 1 divided by
the absolute value of x to the k is
going to be the absolute value of x.
We're going to rewrite this bottom
as 2 times k + 2 factorial.
So now we have the absolute value of x
times the limit as k goes to infinity.
This top 2k factorial
is 1 times 2 times 3,
etc., to 2k and the bottom is 1 times 2
times 3 etc., up to 2k times 2k+1 times
2k+2.
So everything will cancel out up top and
on bottom we're just going to be
left with 2k + 1 times 2k + 2.
So now we have, this equals the absolute
value of x times the limit
as k goes to infinity
of 1 divided by 2k + 1 times 2k + 2.
And this equals 0 and
that is less than 1 for all x.
So that means that our radius
of convergence is infinity
because we're converging for all x.
And our interval of convergence
is negative infinity to infinity.
So we are converging for all values of x.
