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Question
Write the value of \[\lim_{n \to \infty} \frac{1 + 2 + 3 + . . . + n}{n^2} .\]
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Solution
\[\lim_{n \to \infty} \left( \frac{1 + 2 + 3 . . . n}{n^2} \right)\]
\[ = \lim_{n \to \infty} \left( \frac{n\left( n + 1 \right)}{2 n^2} \right)\]
\[ = \lim_{n \to \infty} \left( \frac{\left( n + 1 \right)}{2n} \right)\]
\[ = \frac{1}{2} \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)\]
\[ = \frac{1}{2}\]
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