If $ x \in \mathbb R^n$ with $\|x\|_\infty < 1$, how to make sense of this quantity $W(x):= \sum_{p=1}^\infty\|x\|_p^p$?

Question

In some calculations of mine, I've stumbled on the following object, and I'm wondering if its a something recognisable.

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For $x \in \mathbb R^n$, let $\|x\|_\infty := \max_{1 \le j \le n}|x_j|$, and for $p \in [1,\infty)$, define $\|x\|_p := (\sum_{j=1}^n |x_j|^p)^{1/p}$. Suppose $\|x\|_\infty < 1$ and define $$ W(x):= \|x\|_1 + \|x\|_2^2+\|x\|_3^3+\|x\|_4^4+\ldots = \sum_{j=1}^n\frac{|x_j|}{1-|x_j|}. $$

Question. What is known of the function $W$ ? Does the quantity $W(x)$ have any particular meaning ? Has it been studied elsewhere ? Are there any interesting inequalities, formulae, etc. involving it ?