00:01
For this problem, we are to find a power series representation for f of x, it's equal to 4 over the product of x squared plus 4 and x squared minus 1 using partial fraction decomposition.
00:12
We're given a hint here.
00:13
It says that we have to write 4 over x squared plus 4 times x squared minus 1 as a sum of a over x squared plus 4 and b over x squared minus 1.
00:26
Suppose 4 over x squared plus 4 times x squared minus 1 equals a over x squared plus 4 plus b over x squared minus 1.
00:38
To solve for a and b, we will multiply this by the lcd x squared plus 4 times x squared minus 1.
00:46
From here we have 4 equal to a times x squared minus 1 plus b times x squared plus 4.
00:54
And then that's fixed values for x.
00:57
Let's say x is equal to 0.
01:00
Then we have 4 equal to a times 0 minus 1.
01:06
That's negative 1 plus b times 0 plus 4.
01:11
That's 4 gives us a equal to 4b minus a.
01:17
Let's say x is equal to 1...