For fixed $k$ suppose we have $X_1,...,X_k$ non-negative random variables with density functions.

Setting a): We know $\mathbb{E}[(X_1^n+...+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{N}$.

Setting b): We can approximate $\mathbb{E}[(X_1^n+...+X_k^n)^m]$ up to some additive error $R_{m,n}$ with $\sup\limits_{m \in \mathbb{N}} |R_{m,n}| \cdot \mathbb{E}[(X_1^n+...+X_k^n)^m]^{-1} \xrightarrow{n \rightarrow \infty} 0$.

Is there a nice way to extract the moments $\mathbb{E}[\max(X_1,...,X_k)^a]$?

There is some trickery one can do with the series expansion of the $n$-th root \begin{align*} & (1+x)^{\frac{a}{n}} = \sum\limits_{r=0}^\infty \frac{\prod\limits_{l=0}^{r-1} (a-ln)}{r! n^r} x^r \ , \end{align*} since the convergence $\mathbb{E}[(X_1^n+...+X_k^n)^{\frac{a}{n}}] \xrightarrow{n \rightarrow \infty} \mathbb{E}[\max(X_1,...,X_k)^a]$ 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! Setting (a) is fine, but I am also interested in Setting (b).