Absolute convergence of log function

**cubs3205** · November 26th 2008, 06:25 AM

How should I show the absolute convergence of $\sum \log(1 + a_n)$ ?
I can't think of a test to use for a log function
Thanks!

**Mathstud28** · November 26th 2008, 06:33 AM

Originally Posted by cubs3205

How should I show the absolute convergence of $\sum \log(1 + a_n)$ ?
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 $|a_n|$ converges? If so just replace $a_n$ by $|a_n|$ in the following, it still holds. First we must have the convergence of $\sum{a_n}$ . If that is true, then by definition $\lim_{n\to\infty}a_n=0$ . So using the limit comparison test we have $\lim_{n\to\infty}\frac{\log(1+a_n)}{a_n}$ . Now letting $\varphi=a_n$ it is clear that since $a_n$ is a null sequence (it covnerges to zero) that As $n\to\infty\implies\varphi\to{0}$ and we are left with $\lim_{\varphi\to{0}}\frac{\ln(1+\varphi)}{\varphi} =1$

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