In this paper, Ramanujan studies the convolution sum of divisor functions, which he denotes as $$\sum_{r,s}(n) := \sum_{m = 0}^n \sigma_r(m) \sigma_s(n-m),$$ where above, he defines $\sigma_s(0) = \frac{1}{2}\zeta(-s)$. He then makes the following sequence of limiting arguments, which I did not understand, and was hoping that someone here could help explain:
$$\begin{equation*} \begin{split} \lim_{n \rightarrow \infty} \frac{\sum_{r,s} (n)}{\sigma_{r+s+1}(n)} &= \lim_{n \rightarrow \infty} \frac{\sum_{r,s}(1) + \sum_{r,s}(2) + \cdots + \sum_{r,s} (n)}{\sigma_{r+s+1}(1) + \sigma_{r+s+1}(2) + \cdots + \sigma_{r+s+1}(n)} \\&= \lim_{x\rightarrow 1} \frac{\sum_{r,s}(0) + \sum_{r,s}(1)x + \sum_{r,s} (2)x^2 + \cdots}{\sigma_{r+s+1}(0) + \sigma_{r+s+1}(1)x + \sigma_{r+s+1}(2)x^2 + \cdots } \\ &= \lim_{x\rightarrow 1} \frac{S_r S_s}{S_{r+s+1}}, \end{split} \end{equation*}$$ where above $$S_r = \frac{1}{2}\zeta(-r) + \frac{1^rx}{1-x} + \frac{2^r x^2}{1-x^2} + \frac{3^r x^3}{1-x^3} + \cdots.$$
I didn't follow the reasoning that supports the equivalences above, and so was hoping that someone might be able to clarify the argument(s)? Thanks!