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.