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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