Property of the remainder term in the Euler-Maclaurin formula for $\sum_{i=1}^n\log i$.

Question

From https://en.wikipedia.org/wiki/Stirling's_approximation, I'm having trouble understanding $$R_{m,n}=\lim_{n\to\infty} R_{m,n}+O(n^{-m}).$$ I worked out $$R_{m,n}=\int_1^{n}\frac{P_m(x)}{mx^m}~ dx$$ where $P_m(x)$ is a periodic Bernoulli polynomial, so $$\begin{align} \left|\lim_{n\to\infty} R_{m,n}-R_{m,n}\right|&=\left|\int_n^{\infty}\frac{P_m(x)}{mx^m}~ dx\right|\\ &\le\int_n^{\infty}\frac{|P_m(x)|}{mx^m}~ dx\\ &\le Mn^{1-m} \end{align}$$ so I get $O(n^{1-m})$ which isn't desired.