Taylor

[LostInSpace]

I am a bit confused about taylor approximation. Taylor around x_0 yields

f(x) = f(x_0) + f'(x_0)(x-x_0) + O(x^2)


which is the tangent of f in x_0, where

f'(x) = f'(x_0) + f''(x_0)(x-x_0) + O(x^2)


which adds up to

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)

But it should be

f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + O(x^3)


Where does the 2! come from? Is this approach completely incorrect?

[arildno]

You have ignored in line 3 the O(x^(2))-term from the expansion of f(x).
Hence line 3 is not accurate to O(x^(3)), it's only accurate to O(x^(2)).

[Hurkyl]

Do you remember how to derive, from the limit definition of the derivative, the differential approximation formula:


f(x+\epsilon ) = f(x) + \epsilon f'(x) + \epsilon \delta(x, \epsilon)


Where \lim_{\epsilon \rightarrow 0} \delta(x, \epsilon) = 0?

Try writing the second derivative with limits, and see if any approach suggests itself.