Find power series representations centered at 0 for the...

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x) = ln √(1 - x^2)

Transcript

00:02 Okay, so what is important to note here is that this function f of x is equal to the natural log of 1 minus x squared.

00:17 Okay, well, there's a common taylor series that is easy to know is that so let's do it this way is natural log of 1 plus x is the sum from n equals 1 to infinity of minus 1 to the n plus 1 times x to the n over n factorial or n n.

01:12 Yeah.

01:14 Okay, so this is a common taylor series and well, you can see how to sort of write it in this way is to note that this first function is the same thing as natural log of 1 plus minus x squared, right? so all you have to do is we need to plug in minus x squared for everywhere we see x in here so f of x is equal to the sum from n equals 1 to infinity of minus 1 to the n plus 1 and i'll also put the n here and then we want minus x squared to the n.

01:58 Okay, so now let's simplify this a bit.

02:00 So this is going to be sum from n equals 1 to infinity of minus 1 to the n plus 1 over n and then the minus x squared we can write as a minus 1 to the n and then x to the 2n.

02:17 Okay, so now note that this infinity of minus 1 to the 2n plus 1 over n x to the 2n but minus 1 to the 2n plus 1, 2n plus 1 would always odd because 2n is even and then you add 1 so it's odd and minus 1 to any odd power like minus 1 to the 1 minus 1 to the 3 any odd power is always minus 1.

02:48 So this tells you that this is just equal to minus times sum from n equals 1 to infinity of x to the 2n over n and this is the answer and oh and then we need to find the interval of convergence.

03:12 So what is the interval of convergence for the standard one? well, okay, we can just find the interval of convergence to the standard tricks.

03:31 So let's write this as x to the 2n over n.

03:38 So this is our a n and let's let's actually compute the interval of convergence.

03:48 So the interval of convergence is going we need to look at a n plus 1 over a n.

03:56 So this is going to be x to the 2n 2 times n plus 1 2 times n plus 1 over n plus 1 and then we have to divide by x to the 2n over n...

Question

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x) = ln √(1 - x^2)

Please give Ace some feedback