First slide

Introduction to limits

Question

The value of $lim_{n \to \infty} \frac{1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n (n + 1)}{n^{3}}$ ,is

Moderate

A

1
B

-1
C

$\frac{1}{3}$
D

none of these

Solution

We have,
$\begin{array}{l} lim_{n \to \infty} \frac{1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n (n + 1)}{n^{3}} \\ = lim_{n \to \infty} \frac{\sum_{r = 1}^{n} r (r + 1)}{n^{3}} = lim_{n \to \infty} \frac{\sum_{r = 1}^{n} r^{2} + \sum_{r = 1}^{n} r}{n^{3}} \\ = lim_{n \to \infty} \frac{\frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2}}{n^{3}} \\ = lim_{n \to \infty} \frac{n (n + 1) (n + 2)}{3 n^{3}} = \frac{1}{3} \end{array}$

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