Prove that $\ln(x)$ diverges

Question

Prove that $\ln(x)$ diverges using the fact that the harmonic series diverges.

How can I compare the $\ln$ with the harmonic series, if the harmonic series appears to be more relevant to the derivative of $\ln$?

Edit: show $\ln(x) \rightarrow\infty$ as $x \rightarrow \infty$.

Answer 1

Hint: You can use the integral test.

$\sum \frac{1}{n}$ diverges $\iff$ $\int_1^\infty \frac{1}{x} dx$ diverges.

Well, we know the harmonic series $\sum \frac{1}{n}$ diverges, so the integral must diverge. Evaluate it! :)

Answer 2

We know that $\int^a_1\dfrac{\mathrm{d}t}{t}=\ln(a)$, and hence, we can use the integral test: $$\sum\dfrac{1}{n}\text{ diverges}\iff\int^\infty_1\dfrac{\mathrm{d}t}{t}\text{ diverges}\\ \sum\dfrac{1}{n}\text{ does diverge, hence }\int^\infty_1\dfrac{\mathrm{d}t}{t}=\lim_{x\to\infty}\ln(x)\text{ diverges.}$$