The series $\displaystyle\sum_{n=1}^\infty a_n$ is absolutely convergent if $\displaystyle\sum_{n=1}^\infty |a_n|$ converges. If $\displaystyle\sum_{n=1}^\infty a_n$ converges, but it is not absolutely convergent, then it is conditionally convergent.

If a series is absolutely convergent, then it is also convergent.

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Does the series $\displaystyle\sum_{n=1}^\infty\frac{\sin(4n)}{4^n}$ converge?

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Let’s create a different series by taking the absolute value of each term:

$$ \displaystyle\sum_{n=1}^\infty\frac{|\sin(4n)|}{4^n} $$

Note that $0 \le |\sin(x)| \le 1$ for any $x$, so we can use the direct comparison test here.

$$ \displaystyle\sum_{n=1}^\infty\frac{|\sin(4n)|}{4^n} \le \displaystyle\sum_{n=1}^\infty\frac{1}{4^n} $$

Since the latter series converges (since it’s a geometric series with $|r|<1$), then the original series converges absolutely. Therefore, $\displaystyle\sum_{n=1}^\infty\frac{\sin(4n)}{4^n}$ converges.