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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?

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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?)

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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)

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$a_n=\dfrac{1}{n+n \cos{n} }$ 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 ?

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