# Determine whether the integral $$\int_{1}^{∞}\left ( \frac{1}{x} \right )dx$$  is convergent or divergent

Solution

Given integral is

$$I = \int_{1}^{∞}\left ( \frac{1}{x} \right )dx$$

Clearly the given integrated function $$\left ( \frac{1}{x} \right )$$ is continuous on [1, ∞). So

$$I = \lim_{a \to ∞} \int_{1}^a \left ( \frac{1}{x} \right ) dx$$

$$= \lim_{a \to ∞^-}\left [ ln(x) \right ]_1^a$$

$$= \lim_{a \to ∞^-}\left [ ln(a) – ln(a) \right ]$$

= ln(∞)

= ∞

This means the given integral is divergent.