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Math Help - Convergent or divergent

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    Convergent or divergent

    Is this series convergent or diveregent...and from what method I can prove this.

    <br /> <br />
\sum_{n=1}^{\infty} \frac{cos (n)}{n}<br />
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  2. #2
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    Quote Originally Posted by racewithferrari View Post
    Is this series convergent or diveregent...and from what method I can prove this.

    <br /> <br />
\sum_{n=1}^{\infty} \frac{cos (n)}{n}<br />
    To prove convergence, use Dirichlet's test.

    If you want to know the sum of the series, notice that \sum_{n=1}^\infty\frac{z^n}n = \log(1-z). If z=e^{i\theta} then \log(1-z) = \ln(2\sin\tfrac\theta2) + i(\theta+\pi)/2. Take the real part to see that \sum_{n=1}^\infty\frac{\cos n\theta}n = \ln(2\sin\tfrac\theta2). In particular, if \theta=1 then \sum_{n=1}^\infty\frac{\cos n}n = \ln(2\sin\tfrac12). That argument is not rigorous, because the series for \log(1-z) has radius of convergence 1, so you cannot assume that it behaves well when |z| = 1. However, once you have used the Dirichlet test to prove convergence, the rest of the argument works to give the correct result.
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