00:01
So we want to find the power series representation of three times the natural log of 2 minus x.
00:06
So to do this, let's look at the series, the taylor series around naturalog of 1 minus x.
00:15
Because we can all of, we can rewrite our given here based upon natural log of 1 minus x.
00:24
So this natural 1 minus x, this is a known taylor series of negative sum from minus 1, 2 .2.
00:31
Infinity of x to the n over n.
00:37
And so now we're going to use this for natural log of 2 minus x.
00:50
And how we're going to do that is look at how we can manipulate this to have a 1 minus sum variable here.
00:58
So this is the same as natural log of 2 times 1 minus x over 2.
01:06
So what we're doing is we're factoring out two from each of these terms, right? 2 divided by 2 is 1.
01:12
X over 2 is well, x over 2, but then we have this 2 here.
01:15
And you'll know it works.
01:16
If you distribute the 2 back in, you'll get 2 minus x.
01:19
And this will be useful for us because we could break this up into using the laws of exponents, which is a natural log of 2 plus the natural log of 1 minus x over 2.
01:33
So now we can see natural log of 2 minus x, which is right here.
01:38
Here, natural law of 2 minus x right there, we have this.
01:45
So this, look at that.
01:46
We're doing pretty good, right? this factor of our, what we're looking for is right here, which we can expand like this.
01:54
Natural algorithm 2 is a constant.
01:55
That's nice...