Given power series with coefficients over $\mathbb{N}$, how to find it's singularity and the order of singularities, if it is with natural boundary, how to find it's order of singularities, and compare their orders?
For example, $$S(x)=\sum_{i= 1}^{\infty}\frac{x^i}{(1-x)(1-x^2)\dots(1-x^i)}$$
it is able to be expanded further to a power series with coefficients over $\mathbb{N}$, which has a natural boundary. Then how to find and compare their orders of singularities