Let $p_k(n)$ denote the number of partitions of the positive integer $n$ into at most $k$ parts. I am trying to answer a question which asks me to show that the generating function for the sequence $\{ p_k(n) \}$ is given by $$ P_k(n) := \frac{1}{\prod_{r=1}^k (1-x^r)} $$
Can anyone help me to understand how this might be done?