- #1
LostInSpace
- 21
- 0
I am a bit confused about taylor approximation. Taylor around [tex]x_0[/tex] yields
[tex]
f(x) = f(x_0) + f'(x_0)(x-x_0) + O(x^2)
[/tex]
which is the tangent of f in [tex]x_0[/tex], where
[tex]
f'(x) = f'(x_0) + f''(x_0)(x-x_0) + O(x^2)
[/tex]
which adds up to
[tex]
f(x) &=& f(x_0) + (f'(x_0) + f''(x_0)(x-x_0) + O(x^2))(x-x_0)+O(x^2) \\ &=& f(x_0) + f'(x_0)(x-x_0) + f''(x_0)(x-x_0)^2 + O(x^3)
[/tex]
But it should be
[tex]
f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + O(x^3)
[/tex]
Where does the 2! come from? Is this approach completely incorrect?
[tex]
f(x) = f(x_0) + f'(x_0)(x-x_0) + O(x^2)
[/tex]
which is the tangent of f in [tex]x_0[/tex], where
[tex]
f'(x) = f'(x_0) + f''(x_0)(x-x_0) + O(x^2)
[/tex]
which adds up to
[tex]
f(x) &=& f(x_0) + (f'(x_0) + f''(x_0)(x-x_0) + O(x^2))(x-x_0)+O(x^2) \\ &=& f(x_0) + f'(x_0)(x-x_0) + f''(x_0)(x-x_0)^2 + O(x^3)
[/tex]
But it should be
[tex]
f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + O(x^3)
[/tex]
Where does the 2! come from? Is this approach completely incorrect?