First slide

Introduction to limits

Question

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

Moderate

A

0
B

$- \frac{1}{2}$
C

$\frac{1}{2}$
D

1

Solution

$\begin{array}{l} lim_{n \to \infty} (\frac{1}{1 - n^{2}} + \frac{2}{1 - n^{2}} + \dots + \frac{n}{1 - n^{2}}) \\ = lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{1 - n^{2}} \\ = lim_{n \to \infty} \frac{Σn}{1 - n^{2}} = lim_{n \to \infty} \frac{n (n + 1)}{2 (1 - n^{2})} = lim_{n \to \infty} \frac{1 + \frac{1}{n}}{2 [\frac{1}{n^{2}} - 1]} = - \frac{1}{2} \end{array}$

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