Testing series for divergence

Question

I have the following series:

$$\sum_{n=1}^\infty \ln(2(n+1))- \ln(2n)$$

How can I test it to show it is divergent?

Answer 1

Hint: "Telescoping series." This is of the form $\sum_{n=1}^\infty (a_{n+1}-a_n)$, with $a_n \to \infty$.

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Or, more heavy and complicated: $$ \ln(2(n+1))- \ln(2n) = \ln \frac{n+1}{n} = \ln(1+\frac{1}{n}) \operatorname*{\sim}_{n\to\infty} \frac{1}{n} $$