Question

$\underset{n\to \infty }{\lim }\left[\frac{1+2+3......n-1}{n^2}\right]$

Answer 1

$$\begin{gathered} \Rightarrow \underset{n\to \infty }{\lim }\left(\frac{1+2+3+...n-1}{n^2}\right) \\ \Rightarrow \underset{n\to \infty }{\lim }\left[\frac{n\left(n-1\right)}{2n^2}\right] \\ \Rightarrow \underset{n\to \infty }{\lim }\left[\left(1-\frac{1}{n}\right)\times \frac{1}{2}\right] \\ \mathrm{When}n\to \infty ,\mathrm{then}\frac{1}{n}\to 0. \\ =\frac{1}{2} \end{gathered}$$