I am a bit confused about taylor approximation. Taylor around [tex]x_0[/tex] yields(adsbygoogle = window.adsbygoogle || []).push({});

[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?

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# Confused about taylor approximation

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