I don't get why this is absolutely convergent? $\displaystyle \sum^{\infty}_{n=1} \frac{sin(4n)}{7^n}$
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$\displaystyle \left| a_{n} \right|=\frac{\left| \sin 4n \right|}{7^{n}}\le \frac{1}{7^{n}},$ that's why.
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