Recurring is recurring is recurring is

Algebra Level 4

Let \(f_1^n (x) = \dfrac x{\sqrt[n]{x^n+1}} \) and \( f^p _q (x) = \underbrace{f(f(f(\ldots (f(x)\ldots)}_{q \text{number of times}} \) where \(f^p _1 (x) \dfrac x{\sqrt[p]{x^p+1}} \).

Evaluate \( \displaystyle \lim_{x\to\infty} \sum_{r=1}^n (f^r _r (x))^r \).

The problem is original. Inspiration.
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