How should I show the absolute convergence of $\displaystyle \sum \log(1 + a_n)$?
I can't think of a test to use for a log function
Thanks!
What do you mean by absolute convergence? Do you mean if $\displaystyle |a_n|$ converges? If so just replace $\displaystyle a_n$ by $\displaystyle |a_n|$ in the following, it still holds. First we must have the convergence of $\displaystyle \sum{a_n}$. If that is true, then by definition $\displaystyle \lim_{n\to\infty}a_n=0$. So using the limit comparison test we have $\displaystyle \lim_{n\to\infty}\frac{\log(1+a_n)}{a_n}$. Now letting $\displaystyle \varphi=a_n$ it is clear that since $\displaystyle a_n$ is a null sequence (it covnerges to zero) that As $\displaystyle n\to\infty\implies\varphi\to{0}$ and we are left with $\displaystyle \lim_{\varphi\to{0}}\frac{\ln(1+\varphi)}{\varphi} =1$