# Fractional part always diverge right?

Calculus Level 4

Let $$f_n(x)$$ be an $$n$$-th degree polynomial with all real coefficients and a positive leading coefficient.

For a fixed $$n$$, if $$\displaystyle \lim_{x\to\infty} \left\{ \sqrt[n]{f_n(x)} \right\}$$ converge, what can we say about the leading coefficient of $$f_n$$?

Details and Assumptions:

• $$\{x\}$$ denote the fractional part of $$x$$.

Bonus: Denote the polynomial as $$f_n(x) = a_n x^n +a_{n-1} x^{n-1} + \ldots + a_0$$. Prove that we can find the limit in terms of $$a_n$$ and $$a_{n-1}$$.

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