How would the Dirichlet Test for convergence prove that $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ does in fact converge?

Question

I've been looking for a way to determine whether $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ converges, and the test that I've most often seen recommended seen is the Dirichlet test for convergence.

Reading the theorem for Dirichlet's test, I have to admit that I'm not having much luck applying the test correctly. My initial thought is to find some convergent sequence $b_n$ that is convergent and bounds $|\frac{\cos{n}}{n}|$, though then I'm forced to consider something like $\frac{\cos{n}}{n^2}$, and it's still not clear to me how this would be helpful in proving that $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ converges.

Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps. — robjohn, Apr 23 '19 at 14:51
What are two sequences $a_n,b_n$ that you might combine as $a_n\cdot b_n$ to make the sum term of this series such that the two sequences meet the criteria for Dirichlet's test? — abiessu, Apr 23 '19 at 14:52

score 2 · Accepted Answer · answered Apr 23 '19 at 14:48

2

We have

$$\sum_{n=1}^{\infty}\cos n \dfrac{1}{n}.$$

The $1/n$ terms are monotonically decreasing to zero and the partial sums

$C_N = \sum_{n=1}^N\cos n$ are bounded. Hence, the series is convergent by Dirichlet's test.

answered Apr 23 '19 at 14:48

MachineLearner

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How would the Dirichlet Test for convergence prove that $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ does in fact converge?

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