For fixed $k$ suppose we have $X_1,...,X_k$ non-negative random variables with density functions and we know $\mathbb{E}[X_1^n+...+X_k^n]$ for large $n$. Is there a nice way to extract the moments $\mathbb{E}[\max(X_1,...,X_k)^m]$?

There is some trickery one can do with the series expansion of the $n$-th root \begin{align*} & (1+x)^{\frac{m}{n}} = \sum\limits_{r=0}^\infty \frac{\prod\limits_{l=0}^{r-1} (m-ln)}{r! n^r} x^r \ , \end{align*} since the convergence $\mathbb{E}[(X_1^n+...+X_k^n)^{\frac{m}{n}}] \xrightarrow{n \rightarrow \infty} \mathbb{E}[\max(X_1,...,X_k)^m]$ holds (under mild integrability conditions). However, the calculations become nasty quickly. Is there a better way? For example a nice analytic function $f_n$, which is close enough to the $n$-th root for large values might make the process easier.

Any help is much apprechiated!