수열 합 극한

$n < a _{n} < n + 1$에서 $\displaystyle \sum\limits _{k=1} ^{n} k < \sum\limits _{k=1} ^{n} a _{k} < \sum\limits _{k=1} ^{n} ( k + 1 )$이므로 $\displaystyle \frac{n ( n + 1 )}{2} < \sum\limits _{k=1} ^{n} a _{k} < \frac{n ( n + 3 )}{2}$ $\displaystyle \frac{n ( n + 1 )}{2 n ^{2}} < \frac{1}{n ^{2}} \sum\limits _{k=1} ^{n} a _{k} < \frac{n ( n + 3 )}{2 n ^{2}}$ $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{n ( n + 1 )}{2 n ^{2}} = \lim\limits _{n\rightarrow \infty} \frac{n ( n + 3 )}{2 n ^{2}} = \frac{1}{2}$ $\therefore$ $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{1}{n ^{2}} \sum\limits _{k=1} ^{n} a _{k} = \frac{1}{2}$