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I have $b>a$ and an invertible and infinitely differentiable function $f$. If I want to evaluate $\epsilon=f^{-1}(b-a)$ by writing:
$b\approx a+f'(a)\epsilon+f''(a)\epsilon^2/2+f''(a)\epsilon^3/6+\cdots+f^{(n)}(a)\epsilon^n/n!$
and solving the polynomial equation for $\epsilon$, what is the error incurred?

Paraphrasing, if I call the approximate value of $\epsilon$ evaluated by solving an $n$th order polynomial equation $\epsilon_n$ then is it true that:

$\epsilon=\epsilon_n+O(|b-a|^n$?

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