Can this Non-Linear Residual Function, $R(x)=((x+1)^3-1)^{1/3}-(x+1)$, be expressed exactly using a infinite series (using integer powers in $x$) or a finite polynomial function in $x$?
For example I have found one approximate series / polynomial function that is a reasonably accurate approximation above $x=2$
$$R(x) \approx -\sum_{n=1}^\infty \frac{(-1)^{n-1}}{3 (x+1)^{2n}}=-\frac{1}{3(x^2+2x+2)}$$
Can you please give some advice in how I might proceed with this problem?
Is the method to proceed using the calculus of finite differences to improve the approximation I have obtained or is there some direct transformation I can apply?
Update Added 13/02/2024
Just playing around tonight I've been able to produce a better approximation, i.e.
$$R(x) \approx -\frac{1}{3 (x+1)^2}-\frac{1}{9 (x+1)^5}-\frac{1}{17 (x+1)^8}-\frac{1}{18 (x+1)^{11}}$$
Which is semi-regular but the trial and error iteration technique seems to run out of steam in terms of improving accuracy with increasing number of terms.
Maybe @GregMartin's answer is telling me this, but I'm not sure.
$R(x)$ is not differentiable at $x=0$, so there's no analytic function that will match it exactly. However, $R(x)$ does have a Puiseux series at $x=0$ that will converge for all $x\in\Bbb R$.