First slide

Introduction to definite integration

Question

$The value of lim_{n \to \infty} [\frac{1}{n a} + \frac{1}{n a + 1} + \frac{1}{n a + 2} + \dots + \frac{1}{n b}] is$

Moderate

A

$\log (a b)$
B

$\log (a / b)$
C

$\log (b / a)$
D

$- \log (a b)$

Solution

the given limit
$L = lim_{n \to \infty} [\frac{1}{n a} + \frac{1}{n a + 1} + \frac{1}{n a + 2} + \dots + \frac{1}{n a + n (b - a)}]$
$\begin{array}{l} = lim_{n \to \infty} \sum_{r = 0}^{(b - a) n} \frac{1}{n a + r} \\ = lim_{n \to \infty} \frac{1}{n} \sum_{r = 0}^{(b - a) n} \frac{1}{a + r / n} \\ = \int_{0}^{(b - a)} \frac{d x}{a + x} = [\log (a + x)]_{0}^{b - a} \\ = \log b - \log a = \log (b / a) \end{array}$

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