题目

求下列极限:
(I) $\displaystyle \mathop{\lim }\limits_{n \rightarrow \infty }\left( {\frac{1}{{n}^{2} + n + 1} + \frac{2}{{n}^{2} + n + 2} + \cdots + \frac{n}{{n}^{2} + n + n}}\right)$
(II) $\mathop{\lim }\limits_{n \rightarrow \infty }\left\lbrack {\sqrt{1 + 2 + \cdots + n} - \sqrt{1 + 2 + \cdots + \left( {n - 1}\right) }}\right\rbrack$ ;
(III) $\displaystyle \mathop{\lim }\limits_{n \rightarrow \infty }\mathop{\sum }\limits_{k = 1}^{n}\frac{1}{4{k}^{2} - 1}$
(IV) $\mathop{\lim }\limits_{n \rightarrow \infty }\sqrt[n]{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}}$ ;
(V) $\mathop{\lim }\limits_{n \rightarrow \infty }{\left( \frac{1 + \sqrt[n]{3}}{2}\right) }^{n}.$