Methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$ As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which involves bounding the partial sums of the cosine or sine terms. I have two questions:


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*Are there any other approaches to seeing that these series are convergent? I'm mostly just interested to see what other kinds of arguments might be made.

*What's the best way to show that these two series are only conditionally convergent? I don't even know the textbook approach to that question.


Since you specifically said "Are there any other approaches to seeing that these series are convergent" I will take the bait and give an extremely sketchy argument that isn't meant to be a proof at all.
The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ famously diverges, but very slowly. For large $n$ the numbers being added are so tiny that it takes exponential amounts of time for the millions of tiny terms to amount to much at all.
But $\cos(n)$ will be negative roughly half the time, so there's no possibility of any slow steady accumulation. Thus our sum $\sum_{n=1}^{\infty} \frac{\cos n}{n}$ will remain bounded.
And furthermore, the magnitude of the terms being added gets smaller and smaller, so  in a matter that's roughly analogous to the logic behind the alternating series test, the sum will oscillate [with a period of roughly 6], but with a decreasing amplitude around a value that is an artifact of the first handful of terms.
