So my initial point of confusion is on \begin{equation} \lim_{x\rightarrow\infty} \ x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\cdots \end{equation} which we recognise as \begin{equation} \lim_{x\rightarrow\infty}\ -\text{log}(1-x)=-\infty \end{equation} by properties of the complex logarithm. Firstly how can we resolve our intuition, the sum of positive quantities taken to infinity is negative infinity?
My interest in this question is that I wish to understand why the limit of the polylogarithm, $\text{Li}_{n}(x)$ (of which $\text{Li}_{1}(x)=-\text{log}(1-x)$ ), is -$\infty$ for positive integer $n$ as $x$ goes to infinity. One of my issues is that the branch cut for the polylogarithm is placed on the positive real axis (where I'm interested) in most definitons, can this simply be moved to the negative real axis as for $\text{log}(1-x)$?
HINT: Compute the radius of convergence of the series. Can we consider limit in infinity for this series?