First slide

Introduction to limits

Question

$lim_{x \to 1} \frac{\sum_{r = 1}^{n} x^{r} - n}{x - 1}$ is equal to

Moderate

A

$\frac{n}{x}$
B

$\frac{n (n + 1)}{2}$
C

$1$
D

$0$

Solution

we have,
$\begin{array}{l} lim_{x \to 1} \frac{\sum^{n} x^{n} - n}{x = 1} = lim_{x \to 1} \frac{x - 1}{x - 1} + \frac{x^{2} - 1^{2}}{x - 1} + \frac{x^{3} - 1^{3}}{x - 1} + \dots + \frac{x^{n} - 1^{n}}{x - 1} \\ = 1 + 2 + 3 + \dots + n = \frac{n (n + 1)}{2} \end{array}$

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