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Question
The value of \[\lim_{n \to \infty} \left\{ \frac{1 + 2 + 3 + . . . + n}{n + 2} - \frac{n}{2} \right\}\]
Options
1/2
1
−1
−1/2
Solution
−1/2
\[ = \lim_{n \to \infty} \left[ \frac{n\left( n + 1 \right)}{2\left( n + 2 \right)} - \frac{n}{2} \right]\]
\[ = \lim_{n \to \infty} \frac{n}{2} \left[ \frac{n + 1 - n - 2}{n + 2} \right]\]
\[ = \lim_{n \to \infty} \frac{n}{2}\left( \frac{- 1}{n + 2} \right)\]
\[ = \lim_{n \to \infty} \frac{- 1}{2\left( 1 + \frac{2}{n} \right)}\]
\[ = \frac{- 1}{2\left( 1 + 0 \right)}\]
\[ = - \frac{1}{2}\]
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