I am interested in the asymptotic order of the partition function $p(n)$.
The paper Asymptotic Formulae in Combinatory Analysis proves there are constants $A$,$B$ such that $e^{A\sqrt{n}} < p(n) < e^{B\sqrt{n}}$ by elementary means. Here is the argument for one side of the inequality, it is just the inductive step which I have not understood:
I assume that $p_r(n)$ is defined to be $0$ for negative $n$, then the inequalty (2.22) does not hold for these $n$. In which case the sum $\{n^{s-1} + (n-s-1)^{s-1} + (n-2s-2)^{s-1} + \ldots\}$ must be finite, ending before $n-ks-k$ becomes negative. On the other hand the use of telescoping down to $n^s$ suggests that the sum is infinite, otherwise we would end up with $n^s - (n-ks-k)^{s-1}$ and are unable to throw away the $k$ term.
Thank you to anyone who will help me understand this argument.