Integral of Reciprocal is Divergent/Unbounded Above

Theorem

$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$

Thus the improper integral $\ds \int_1^{\to +\infty} \frac {\d x} x$ does not exist.

Proof 1

From Harmonic Series is Divergent, we have that $\ds \sum_{n \mathop = 1}^\infty \frac 1 n$ diverges to $+\infty$.

Thus from the Cauchy Integral Test:

$\ds \int_1^n \frac {\d x} x \to +\infty$

diverges.

$\blacksquare$

Proof 2

From the definition of natural logarithm:

$\ds \ln x = \int_1^x \dfrac 1 t \rd t$

The result follows from Logarithm Tends to Infinity.

$\blacksquare$