\lim_{n\rightarrow\infty}\frac{n\ln(n+1)-\ln n!}{2n+2}=\lim_{n\rightarrow\infty}\frac{n\ln(n+1)-\ln\sqrt{2\pi n}\left(\frac ne\right)^n}{2n+2}-\lim_{n\rightarrow\infty}\frac{\ln(1+e_n)}{2n+2}