First slide

Introduction to limits

Question

$lim_{n \to \infty} \{\frac{1}{1 - n^{2}} + \frac{2}{1 - n^{2}} + \frac{3}{1 - n^{2}} + \dots + \frac{n}{1 - n^{2}}\}$ is equal to

Moderate

A

0
B

$- 1 / 2$
C

$1 / 2$
D

1

Solution

We have,
$\begin{aligned} lim_{n \to \infty} \{\frac{1}{1 - n^{2}} + \frac{2}{1 - n^{2}} + \frac{3}{1 - n^{2}} + \dots + \frac{n}{1 - n^{2}}\} \\ = lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{1 - n^{2}} = lim_{n \to \infty} \frac{n}{2 (1 - n)} = - \frac{1}{2} \end{aligned}$

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