a.Với: $\forall x \in [k,k+1]\Rightarrow \ln k \leq\ln x \leq \ln (k+1)$
$\Rightarrow \int\limits_{k}^{k+1} \ln k dx \leq \int\limits_{k}^{k+1} \ln x dx\leq\int\limits_{k}^{k+1} \ln (k+1) dx$
$\Rightarrow \ln k \leq \int\limits_{k}^{k+1} \ln x dx\leq \ln (k+1)$
$\Rightarrow$ (ĐPCM)
b.Theo câu a:
$\ln 1+\ln 2+...+\ln (n-1)\leq \int\limits_{1}^{2}\ln x dx+ \int\limits_{2}^{3}\ln x dx+...+ \int\limits_{n-1}^{n}\ln x dx$
$\leq \ln 2+\ln 3+...+\ln n, \forall n\in N$\ $\left\{ \begin{array}{l} \end{array} \right.\left. 0,1 \right \}$
$\Rightarrow \ln [(n-1)!]\leq \int\limits_{1}^{n}\ln x dx\leq \ln[n!]$
$\Rightarrow \ln [(n-1)!]\leq [x\ln \frac{x}{e}]_{1}^{n}\leq \ln[n!] $
$\Rightarrow \ln [(n-1)!]\leq n\ln \frac{n}{e}-\ln \frac{1}{e}\leq \ln[n!]$
$\Rightarrow \ln [(n-1)!]\leq \ln[(\frac{n}{e})^{n}.e]\leq \ln[n!]$
$\Rightarrow (n-1)!\leq (\frac{n}{e})^{n}.e\leq n! $
$\Rightarrow(n-1)!\leq n^{n}.e^{-n}.e\leq n!$
$\Rightarrow$ (ĐPCM)
c.Theo b: $(n-1)!\leq n^{n}.e^{-n}.e\leq n!$
$\Rightarrow \frac{1}{n}n!\leq n^{n}.e^{-n}.e\leq n!$
$\Rightarrow \frac{1}{\sqrt[n]{n}}.\sqrt[n]{n!}\leq n.e^{-1}.\sqrt[n]{e}\leq \sqrt[n]{n!}$
$\Rightarrow \frac{\sqrt[n]{e}}{e}\leq \frac{\sqrt[n]{n!}}{n}\leq \frac{\sqrt[n]{n}.\sqrt[n]{e}}{e} $
Do:$\mathop {\lim }\limits_{n \to +\infty } \frac{\sqrt[n]{e}}{e}=\mathop {\lim }\limits_{n \to +\infty }\frac{\sqrt[n]{n}.\sqrt[n]{e}}{e}=\frac{1}{e}$
$\Rightarrow \mathop {\lim }\limits_{n \to +\infty }\frac{\sqrt[n]{n!}}{n}=\frac{1}{e}$