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

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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.

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