Given series $a_n$ converges absolutely, then how to prove series $\log(1+a_n^4)$ converges

Question

Given series $a_n$ converges absolutely, then how to prove series $\log(1+a_n^4)$ converges.

Which test should I use?

Thanks for helping.

Answer 1

We have from the absolute convergence of the series $\Rightarrow$ $|a_n| \to 0 \Rightarrow |a_n| \leq 1, n > N\Rightarrow |a_n|^4 \leq |a_n|, n > N\Rightarrow \log(1+a_n^4) \leq a_n^4 \leq |a_n|$, and the comparison test can be used to conclude.

Answer 2

Hint 1: if $\sum\limits_{n=1}^\infty a_n$ converges, then $\lim\limits_{n\to\infty}a_n=0$.

Hint 2: $\lim\limits_{x\to0}\frac{\log(1+x)}x=1$.

The Comparison Test should work.