# Does the series $\sum_{n=1}^\infty 1/(n+n \cos(n))$ converge or diverge?

Does the series $$\sum_{n=1}^\infty 1/(n+n \cos(n))$$ converge or diverge? How can I use Direct Comparison Test for this problem?

Notice that $$\sum_{n=1}^{\infty}\frac{1}{n+ncos(n)}≥ \sum_{n=1}^{\infty}\frac{1}{n+n}$$ (Why?)

Now:

$$\sum_{n=1}^{\infty}\frac{1}{2n}=\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n}$$ diverges (Why?)

Observe that

$$n + n \cos n \leq 2n$$

from which one easily sees the divergence of the series after taking reciprocals. (notice tha Harmonic $$\sum \frac{1}{2n}$$ diverge)

$$a_n=\dfrac{1}{n+n \cos{n} }$$ and let $$b_n=\dfrac{1}{n}$$

$$\lim_{n \to \infty}\dfrac{a_n}{b_n}=\dfrac{1}{1+cosn}=finite$$

$$\sum_n a_n$$ and $$\sum_n b_n$$ converge or diverge together.$$\sum b_n$$ is a divergent or convergent series ?