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I've been stuck with this problem for a couple of days trying to solve it but got no where till now. The problem states that we have to prove if the series given below is convergent or divergent, if possible use the limit comparison test.

$$\sum_{n=0}^\infty \frac{1+\sin n}{10^n}$$

I'm honestly stuck with this, I managed to pass the limit comparison test by assuming $b_n=\frac{n}{10^n}$ but then stuck with trying to find if the series $b_n$ is convergent. Can someone please help me out? Thanks!

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Use the fact that if a series absolutely converges then it converges, and use the fact that you have:

$$\left|\frac{1+\sin n}{10^n}\right|\leq \frac{2}{10^n}$$

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HINT: Both $$\sum_{n=1}^\infty \frac{1+1}{10^n}$$ and $$\sum_{n=1}^\infty \frac{1-1}{10^n}$$ converge. This is applicable because $-1\leqslant\sin x\leqslant1$

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