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As you can guess from the tittle, just to satisfy my curiosity, I've been looking for series representations of powers of logarithms, something formally expressed as $$ \ln^{m}(n)=\sum_{...}^{\infty}\cdots $$ where $n\ge2 \;\wedge\;n \in \mathbb{N}$.

So thats it. Have you ever found something like this? Could you post it here or point to reffreneces?

Thanks.

EDIT:

I think that I was not very clear in my question, So I'm asking for a series valid for $n\ge 2\;\wedge\;n \in \mathbb{N} $.

Neves
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From this we have:

$$\ln ^k(x)=\sum _{n=k}^{\infty } \frac{(-1)^n (1-x)^n k! S_n^{(k)}}{n!}$$

where: $ S_n^{(k)}$ is Stirling number of the first kind.

Mariusz Iwaniuk
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  • Unfortunately this only converges for $|z|<1$ (see the wikipedia page). So it is not valid for $n>1$ as I asked for. – Neves Dec 09 '19 at 09:27
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    @Neves set $x=1/q$ and note that $\ln^k(1/q)=(-1)^k\ln^k(q)$. Since $q>1$ we have $|1/q|<1$ and the convergence of the series. – clathratus Dec 09 '19 at 23:46