## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\ldots+\frac{n-1}{n^{2}}\right)$
Solution

$\lim _{n \rightarrow \infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\ldots+\frac{n-1}{n^{2}}\right)=\lim _{n \rightarrow \infty}\left(\frac{1+2+3+\ldots+(n-1)}{n^{2}}\right)=$

$=\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left(\frac{n(n-1)}{2}\right)=\lim _{n \rightarrow \infty} \frac{n-1}{2 n}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}(n-1)}{\frac{1}{n} 2 n}=\lim _{n \rightarrow \infty} \frac{1-\frac{1}{n}}{2}=$

$=\frac{1-0}{2}=\frac{1}{2}$

## Problem Kuznetsov Limits 6-1