# Does the series $\sum_{n=1}^{\infty} \frac{\sin{6n}}{1+2^n}$ converge or diverge?

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

• 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. – DiegoMath Apr 17 at 22:39
• Thus, the series converges absolutely. Now, by comparison test... – DiegoMath Apr 17 at 22:39
• @DonAntonio Indeed, thanks! I fixed the comment – DiegoMath Apr 17 at 22:44
• So by taking the absolute value of the original expression, does this mean that the series is absolutely convergent? – Jeremiah Apr 17 at 22:49
• @Jeremiah Indeed so, and thus it also converges. – DonAntonio Apr 17 at 23:59

We know that

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

converges.

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.