00:01
So we have this function, f of x is 1 over 1 plus x squared.
00:04
We want to find the taylor series expansion around the point x equals 0.
00:08
So remember the taylor series expansion is this series, n equals 0 to infinity.
00:14
We take the nth derivative, evaluate that 0, divide by n factorial times by x to the n.
00:20
So the only thing we need to find are these terms, the nth derivative and evaluate them at 0.
00:25
So first, if n is 0, this is just f of 0, which is 1 over 1.
00:30
1 plus 0 squared, which is 1 over 1, which is 1.
00:34
So f of 0 is 1.
00:36
Now we need to take some derivatives.
00:38
F prime of x is going to be minus 2 over 1 plus x cubed.
00:45
So f prime of 0 is going to be minus 2.
00:51
The second derivative, f prime prime of x, is going to be minus 2.
00:56
So we need to take a minus 3 here.
00:59
So we're going to get 6 over 1 plus x to the 4, right? so the second derivative at 0 is 6.
01:13
The third derivative of x is going to be, so we're going to get a minus 4 here when we differentiate.
01:21
So minus 4 times 6 is minus 24, divided by 1 plus x to the power 5...