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Question
\[\lim_{n \to \infty} \left[ \frac{1 + 2 + 3 . . . . . . n - 1}{n^2} \right]\]
Solution
\[\Rightarrow \lim_{n \to \infty} \left( \frac{1 + 2 + 3 + . . . n - 1}{n^2} \right)\]
\[ \Rightarrow \lim_{n \to \infty} \left[ \frac{n\left( n - 1 \right)}{2 n^2} \right]\]
\[ \Rightarrow \lim_{n \to \infty} \left[ \left( 1 - \frac{1}{n} \right) \times \frac{1}{2} \right]\]
\[When n \to \infty , then \frac{1}{n} \to 0 . \]
\[ = \frac{1}{2}\]
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