Good morning fellow mathematicians welcome back to my video. Today we are going to talk about the radius, the radio hehe,
of
convergence of the cos(x) Taylor series expansion and the sin(x) Taylor series expansion. Two radios in
one video. Let's go in and get started just like with the exponential function
we are, well, just going to make use of the ratio test you find in your
Anal I (lol) notes
so you see a
series is nothing but the infinite summation of sequences
so those are our a_n's and
the ratio test just tells us that we have the limit as n approaches infinity of the absolute value of a_(n+1)
over a_n and if this thing is strictly less than 1, well this series converges absolutely
for all the given values of x that we plug in here. We will see what those values of x are actually. This time
I'm not going to make a mistake, hopefully. Last time, I said something
correctly, but I've written something wrong on the blackboard. This happens from time to time, I'm terribly sorry for this.
So let's go ahead and get started with the cosine at first.
I'm going to refer to this limit right here as just capital L,
it's easier to write.
So we are going to take the limit of the absolute value of
This is going to be quite a mess. Plug the n+1 into here at first so: -1 to the (n+1)-th
power, x to the 2n+
2 times n+1 power, so don't make a mistake here. You have to plug n+1 into here, so
Separately over
2 times n+1
factorial.
Ok, you see? This is how it works. Over this term basically itself. So -1 to the n-th power
x to the 2n power over 2n factorial.
Ok taking the absolute value on everything. This is looking so good as just a huge fraction right here. Ok, so you see this?
-1 to the n-th power and this -1 to the n-th power would cancel out to -1.
But since we're taking the absolute values, it's multiplicative and we can take the absolute value of this -1.
Meaning we are going to get rid of this -1 overall because absolute value of -1 is just +1.
Okay, with that out of the way we get rid of this right here.
Why not distribute the 2 into everything on here and let's take the reciprocal here. So we are ending up with the limit of
absolute value
x to the 2 times n + 2.
Again, we're going to take the reciprocal meaning we're going to bring this up to the top; times 2n factorial.
Over, okay down here we have x to the 2n-th power and then we have, well,
times 2n+2 factorial
You see this is what we are going to get right now.
If we take a closer look this x to the 2n-th power is going to cancel out, so we have x squared up here and
down here. What is 2n+1 factorial? This is nothing but 2n-
2n+2 factorial: this is nothing but 2n+2 times 2n+1 times 2n factorial
So if you would write this all out to 2n+2 times 2n+1 times 2n factorial,
you're going to see that this and that is going to cancel out in the end to x squared over this chunk right here.
So we are going to take the limit of the absolute value of x squared over
2n+2
times 2n+1 and don't forget we are going to take the limit as n approaches infinity. You can distribute this absolute value
into the numerator and denominator. In the numerator is just going to stay as absolute value of x squared but down here
Those are all natural numbers. That's the
property of the
factorial so they absolute value of this chunk is just the chunk itself down here. And if we take the limit as n
approaches infinity. This is some finite number
over infinite here
basically goes to 0. So it doesn't matter which value of x we take this thing is going to 0 for n
to infinity.
Meaning our radius of convergence
is infinite in this case, so it doesn't matter what complex or real value you plug in for your x right here,
anyway it doesn't matter it goes to zero in the limit. And now we can do the same spiel for our sine up here.
Let me write everything else. [MY EARS]
So I've written everything out and you see it's even more of a mess just because of this n+1 up here.
So once again, we can just factor everything out and take the reciprocal of this fraction down here.
So getting rid of the complex fraction. Once again our limit as n approaches infinity is going to be referred of as capital L.
So we have capital L of. Up here, we have -1 to the  (n+1)-th power.
Then we have x to the 2n
Plus 1 plus 2 power. I'm going to put it this way. Times
2n+1 factorial over
what do we have down here now? So we have 2n
plus 2 plus 1
factorial times -1 to the n-th power and
then we have x to the (2n+1)-th power
Taking the absolute value over everything, hehe,
this is quite a mess
like I said.
You see, just like before this -1 is going to vanish in the end because of the absolute value. This is good.
Ok, and you see this x to the (2n+1)-th power is going to cancel out
up here, that's why I've written it that way. So we have x squared up here. And what is this chunk right here?
So this basically is just
2n+3 factorial, so just like before we can rewrite this factorial as
2n+3 times 2n+2 times
2n+1 factorial.
And you see once again this and that is going to cancel out to a certain limit, namely the limit of
x squared over, once again those are just
integers basically. Natural numbers at that, not integers, positive integers. Times 2n+3
2n+2. And you see once again, it doesn't matter which value for x you plug into here. In the limit
this is going to zero as n approaches
infinity. Meaning equivalently that our radius of convergence is once again infinity.
This basically has to do with the fact that we can express cosine and sine with respect to the exponential function with
which also has a radius of infinity right here. I hope this video helped a little bit.
If it did,
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