real analysis - $sumlimits_{n=1}^infty log(1+a_n)$ converges absolutely $iffsumlimits_{n=1}^infty a_n$ converges absolutely.

$$\sum\limits_{n=1}^\infty \log(1+a_n) \text{ converges absolutely} \Leftrightarrow \sum_{n=1}^\infty a_n \text{ converges absolutely}.$$

How to prove this,

Suppose $$\sum_{n=1}^\infty a_n \text{ converges absolutely}.$$ Let $u_{n}=a_{n}$ and $v_{n}=\log(1+a_n)$, then $$\lim_{n\to\infty} \frac{u_{n}}{v_{n}}=1>0 \implies\sum_{n=1}^\infty \log(1+ a_n) \text{ converges absolutely}.$$ How to prove the converse part?

Answer

Hint: From the definition of $\ln'(1),$ we have

$$\lim_{u\to 0}\frac{\ln (1+u)}{u} = 1.$$

Thus there is $a>0$ such that

$$\frac{1}{2}\le \left|\frac{\ln (1+u)}{u}\right| \le \frac{3}{2}$$

for $u\in (-a,a),u\ne0.$