Let p(n,m) be the number of partitions of an integer n into integers <= m, we have a well-known asymptotic expression:

For a fixed m and n-> infinity, p(n,m)=n^(m-1)/(m!(m-1)!) (1+O(1/n))

My question is: why the error O(1/n) is indepdent of m? Or how can it be extended for m growing slowly with n? Please help me to find the answer or the references. Thanks.