In this video, we're gonna
to find the interval of
convergence for the
following power series.
So what we do is we apply the ratio test
and as a reminder, the ratio test is A.
We're using K as a subscript in this one.
A sub K plus one, over A sub K.
But typically what I do,
I used K, so let's put a K there.
Instead of instead of doing A sub K
plus one over A sub K,
I'll do A sub K plus one.
So X minus one, to the K plus one,
over three times K plus one.
Putting that K plus one in
parentheses is important.
Because a lot of times what
I'll see is people will
write it is just three K plus one.
At the end, that three
needs to be distributed,
and I'm just gonna multiply
by one over A sub K
or the reciprocal of A sub K,
which is three K over
X minus one to the K.
We have our limit setup,
the rest is just doing some
simplification and some algebra.
So we have the limit as K goes to
infinity, of the absolute value.
Now we're gonna use the
property of exponents here.
This is gonna become A, or excuse me,
X minus one to the K, times X minus one.
All right.
This is just a property of exponents
and we have a three K here.
All over, three K plus three.
So put that in parentheses, times
X minus one to the K and now we look
for anything that'll cancel.
These two terms will cancel
and what I see now is this term
no longer depends on K.
So I am gonna pull that out and when I do,
I do need to maintain that absolute value.
So I've got the absolute
value of X minus one
and now the rest does.
Limit SK goes to infinity
and I do not need
the absolute value on this part,
because it's all positive values.
Three K, over three K, plus three.
At this point you're probably
pretty comfortable with this limit.
But if you're not, you could do
a round of L'Hospital's Rule.
This limit goes to one.
So we get the absolute value of
X minus one is less than one.
Now if all we're looking for is the
radius of convergence, we have it already.
It's equal to one.
But we were asked to find
the interval of convergence.
So we have negative one
is less than X minus one,
is less than one and we'll
add one to all three parts.
So we'll get X has to be
in between zero and two.
Now comes the very important part,
that we need to do for the
interval of convergence,
that we didn't need to do for the radius,
which is always half of
your interval anyways,
is we have to test the endpoints.
Which means, we have to take
each of these endpoints,
plug them into the original, and see if
that series converges or not.
You plug them in for X.
So the first one's gonna be X equals 0,
which will give us the series,
from K equals one to infinity,
of zero minus one.
So negative one to the K, over three K.
I'm not gonna go through all the steps.
But this is convergent by
the alternating series test.
Next one we're gonna
test is for X equals two.
So we have K equals one to infinity.
We're plugging two in for X here.
Two minus one is one.
So I have one to the K, over three K.
One to the K is just one.
So I have K equals one to infinity,
of one over three K.
Well this is one third,
times our harmonic series.
So this is divergent.
You could either say by P series,
or you could say because
it's the harmonic series.
It's a very famous series.
So that means we do want to
include zero in our interval.
But do not include two.
So our interval of convergence,
is going to be zero to two,
for this particular series.
Let's put this up here,
is zero to two including
zero and not including two.
