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Question
Evaluate: \[\lim_{n \to \infty} \frac{1 . 2 + 2 . 3 + 3 . 4 + . . . + n\left( n + 1 \right)}{n^3}\]
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Solution
\[\lim_{n \to \infty} \frac{1 . 2 + 2 . 3 + 3 . 4 + . . . + n\left( n + 1 \right)}{n^3}\]
\[ = \lim_{n \to \infty} \frac{\sum^n_{k = 1} k\left( k + 1 \right)}{n^3}\]
\[ = \lim_{n \to \infty} \frac{\sum^n_{k = 1} k^2 + \sum^n_{k = 1} k}{n^3}\]
\[ = \lim_{n \to \infty} \frac{\frac{n\left( n + 1 \right)\left( 2n + 1 \right)}{6} + \frac{n\left( n + 1 \right)}{2}}{n^3}\]
\[ = \lim_{n \to \infty} \frac{2n\left( n + 1 \right)\left( n + 2 \right)}{6 n^3}\]
\[ = \frac{1}{3} \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)\left( 1 + \frac{2}{n} \right)\]
\[ = \frac{1}{3} \times \left( 1 + 0 \right) \times \left( 1 + 0 \right) \left( \lim_{n \to \infty} \frac{1}{n} = 0 \right)\]
\[ = \frac{1}{3}\]
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