The given function is:

$f_n(x) = \left(\frac{x^n}{2n + 1}\right)^{\frac{1}{n}}$

where $x > 0$.

Would you like me to:

1. Analyze the behavior of $f_n(x)$ as $n \to \infty$?
2. Differentiate $f_n(x)$ with respect to $x$?
3. Explore the limits of $f_n(x)$ for specific values of $n$?
4. Investigate properties like concavity/convexity or critical points?
5. Simplify or further manipulate this function?

Feel free to ask for any details or explanations!

Tip: When dealing with functions defined in terms of a parameter $n$, analyzing their behavior for large $n$ can provide useful insights into their long-term behavior or asymptotic properties.