数学试题讲解

设

a_{n} = \sum_{i = 1}^{n} \sin \frac{i}{n^{2}}, b_{n} = \sum_{i = 1}^{n} \frac{i}{n^{2}}, c_{n} = \sum_{i = 1}^{n} \tan \frac{i}{n^{c}}

.
(1) 证明:

a_{n} < b_{n} < c_{n}

；
(2) 求

lim_{n \to \infty} \frac{a_{n}}{c_{n}}

;
(3) 求

lim_{n \to \infty} a_{n}