Number of partitions of an integer is subexponential

Question

Recently I came across the Hardy-Ramanujan theorem, which states that the number of partitions of an integer $n$ denoted by $p(n)$ behaves asymptotically like $$e^{\pi\sqrt{2n/3}}.$$ This implies that $$\limsup p(n)^{1/n}=1.$$ I was wondering if it is relatively easier to derive the last statement or one has to go through the proof of Hardy and Ramanujan.

Answer 1

$$f(z)=\sum_{n\geq 0}p(n) z^n = \prod_{k\geq 1}\frac{1}{1-x^k} $$ is a holomorphic function in the region $|z|<1$ and the statement $$ \limsup p(n)^{1/n}=1$$ is equivalent to the statement "the radius of convergence of $f(z)$ at the origin is one", which is trivial.