First slide

Introduction to limits

Question

If the value of $lim_{x \to 1} \frac{x + x^{2} + \dots + x^{n} - n}{x - 1}$ , is

Moderate

A

$n$
B

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

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

$\frac{n (n - 1)}{2}$

Solution

We have,
$lim_{z \to 1} \frac{x + x^{2} + x^{3} + \dots + x^{n} - n}{x - 1}$
$= lim_{x \to 1} \frac{(x - 1) + (x^{2} - 1^{2}) + (x^{3} - 1^{3}) + \dots . . + (x^{n} - 1^{n})}{x - 1}$
$= lim_{x \to 1} \{\frac{x - 1}{x - 1} + \frac{x^{2} - 1^{2}}{x - 1} + \frac{x^{3} - 1^{3}}{x - 1} + \dots + \frac{x^{n} - 1^{n}}{x - 1}\}$
$= 1 + 2 (1)^{2 - 1} + 3 (1)^{3 - 1} + \dots . . + n (1)^{n - 1}$
$= 1 + 2 + 3 + \dots \dots + n = \frac{n (n + 1)}{2}$

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