TITLE: Compute $\lim_{n\to \infty} \left({1 \over n^2} + {2 \over n^2} + \cdots + {n - 1 \over n^2}\right)$
QUESTION [5 upvotes]: $$\lim_{n\to \infty} \left({1 \over n^2} + {2 \over n^2} + \cdots + {n - 1 \over n^2}\right)$$
I tried solving this by finding $d$ which is $a_2 - a_1 = d$, but I don't know how to continue with it because it goes to inifnity and $S_n$ i beilieve works only for an infinite set.
I also tried with sandwich, i mean $b_c \le a_n \le c_n$ but when I checked $b_n$ and $c_n$ limits they were not equal to one another so I couldn't find $a_n$ limit.
Would love to get some help, and sorry for my english, I study this on another language.

REPLY [2 votes]: $\lim\limits_{n\to\infty} (\frac{1}{n^2} +\frac{2}{n^2}+\frac{3}{n^2}+...+\frac{n-1}{n^2})= \lim\limits_{n\to\infty} \frac{n(n-1)}{2n^2}=\lim\limits_{n\to\infty} \frac{(1-\frac{1}{n})}{2}=\frac{1}{2}$