real analysis - Limit of $(x_n)$ with $0

Let $0 < x_1 < 1$ and $x_{n + 1} = x_n - x_n^{n + 1}$ for $n \geqslant 1$.

Prove that the limit exists and find the limit in terms of $x_1$.

I have proved the existence but cannot manage the other part.

Thanks for any help.

Answer

You can clearly write each term $x_{n}$ as a polynomial in $x=x_1$, and it should be apparent that all such polynomials are $x + O(x^2)$. Then the difference $x_{n+1}-x_{n}=x_{n}^{n+1}$ is $O(x^{n+1})$, and so the coefficient of $x^{k}$ is the same for all $x_{n}$ with $n\ge k$. This allows us to determine the power series expansion of $x_\infty=\lim_{n\rightarrow\infty}x_{n}$: it is
$$
x_\infty(x) = x-x^2-x^3+2x^4+3x^6-20x^7+30x^8-11x^9-31x^{10}+228x^{11}+\dots
$$
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