I'm having trouble identifying which test to use since the terms in the series oscillate between positive and negative values.

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    $\begingroup$ Note that $$\frac{|\sin 6n|}{1+2^n}\leq\frac{1}{1+2^n}<\frac{1}{2^n}$$ and $\sum_{n=1}^\infty\frac{1}{2^n}$ converges. $\endgroup$ – DiegoMath Apr 17 at 22:39
  • $\begingroup$ Thus, the series converges absolutely. Now, by comparison test... $\endgroup$ – DiegoMath Apr 17 at 22:39
  • $\begingroup$ @DonAntonio Indeed, thanks! I fixed the comment $\endgroup$ – DiegoMath Apr 17 at 22:44
  • $\begingroup$ So by taking the absolute value of the original expression, does this mean that the series is absolutely convergent? $\endgroup$ – Jeremiah Apr 17 at 22:49
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    $\begingroup$ @Jeremiah Indeed so, and thus it also converges. $\endgroup$ – DonAntonio Apr 17 at 23:59

We know that

$$\int_0^{\infty} \dfrac{sin(x)}{x} \ dx$$ and therefor $$\sum_{n=0}^{\infty} \dfrac{sin(n)}{n}$$


There are many detailed proofs on the web so I will not focus on proving that.

But you can see that:

$$|\dfrac{sin(n)}{2^n}| \leq |\dfrac{sin(n)}{n}|$$

And therefor it must be convergent.


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