I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$
I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$
$$n! \leq ne \left(\frac{n}{e} \right)^n$$ and $$n! \geq e \left(\frac{n}{e} \right)^n$$
This is what I have so far:
$$\ln n! = \sum_{k=1}^n \ln k = \sum_{k=2}^n \ln k \geq \int_1^n \ln x \, dx = n\ln n$$
So I have $$\ln n! \geq n\ln n \Rightarrow n! \geq n^n$$
Where do I go next?