Is it true that

$$f(x)=\lim_{n\to\infty} 2 \sum _{k=0}^n \left((k-1) \text{Li}_k\left(\frac{f(x)}{n^2}\right)-x \text{Li}_{k-1}\left(\frac{f(x)}{n^2}\right)\right)?$$

Here, $f(x)$ is an arbitrary function that I tested. I found this by chance, but numerically it looks OK (tried 5000 terms with $\exp$, $\cosh$, $\sinh$ and $\sin$). Is there any justification?

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