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.

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    $\begingroup$ $\cos{n}={e^{in}+e^{-in}\over 2}$ $\endgroup$ – saulspatz Apr 23 at 14:44
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    $\begingroup$ 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. $\endgroup$ – robjohn Apr 23 at 14:51
  • $\begingroup$ 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? $\endgroup$ – abiessu Apr 23 at 14:52
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    $\begingroup$ this answer might prove useful as an example. $\endgroup$ – robjohn Apr 23 at 15:09

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.


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