00:01
Okay, so we have to prove that pn on c is f on c.
00:08
So first, that's the expression of the pn expansion of the taylor polynomial around c.
00:16
And so what we want is replace in all the xc.
00:21
So if we're replacing all the xc, we see that all these terms have c minus c, that is zero.
00:31
Except for the first one, except for the constant term, that would be f of c.
00:37
That's what we had to prove.
00:39
Now, we have to prove that if we evaluate in the k derivative of pn, we obtain the k derivative of f in c.
00:51
So this is the derivative.
00:53
You see that the first term that we have is the k term.
01:02
And this is the expression you can get.
01:05
I recommend if you don't see this expression, you don't think it's true...