# Showing $\lim_{n \to +\infty} \log(n!)/(n\log n) = 1$ without using Stirling approximation

We compute $$\lim_{n\to\infty} \frac{ \log( (n+1)! ) - \log (n!) }{ (n+1)\log(n+1) - n\log n} = \lim_{n\to\infty} \frac{ \log (n+1) }{ \log (n+ \theta_n)+1}=1$$

where $\theta_n \in (0,1)$ is obtained from the Mean Value Theorem. It follows from the Stolz-Cesàro theorem that the limit in question is also equal to $1.$

With an integral test for convergence: $\displaystyle \int_1^n \log(x)dx \leq \sum\limits_{k=2}^n \log(k) = \log(n!) \leq \int_2^{n+1} \log(x)dx$.

You can deduce the stronger result: $\log(n!)=n\log(n)-n + o(n)$.

We have $$\log n!=\sum_{j=1}^n\log\frac jn+n\log n,$$ so we have to show that $\frac 1{n\log n}\sum_{j=1}^n\log\frac jn\to 0.$ We use the inequality $\log(1+t)\geq t-t^2/2$ for $t\geq -1$, to get $$0\geq \frac 1{n\log n}\sum_{j=1}^n\log\frac jn\geq \frac 1{n\log n}\sum_{j=1}^n\left(\frac jn-1\right)-\frac 1{2n\log n}\sum_{j=1}^n\left(\frac{j}{n}-1\right)^2.$$ We have in the RHS $\frac 1{\log n}$ times a Riemann sum (associated with $t\mapsto t-1-\left(t-1\right)^2/2$), hence $\frac 1{\log n}$ times a bounded term, which gives the result.