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Math Help - Absolute Convergence Problem

  1. #1
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    Absolute Convergence Problem

    1.) Determine if the series is absolutely convergent, conditionally convergent or divergent: \sum_{n=1}^{\infty} \frac{cos(n\pi/3)}{n!}

    I think we use the Ratio Test for this one but can someone please clarify the steps I need to take to finish this problem? Thanks!
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  2. #2
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    Here is a strong hint.
    \left| {\frac{{\cos \left( {\frac{{2n\pi }}<br />
{3}} \right)}}<br />
{{n!}}} \right| \leqslant \frac{1}<br />
{{n!}}<br />
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by amp10388 View Post
    1.) Determine if the series is absolutely convergent, conditionally convergent or divergent: \sum_{n=1}^{\infty} \frac{cos(n\pi/3)}{n!}

    I think we use the Ratio Test for this one but can someone please clarify the steps I need to take to finish this problem? Thanks!
    Plato obviously used the right method...but just to reinforce it if you used the ratio test you would have \lim_{n \to {\infty}}\bigg|\frac{cos(\pi(n+1))}{(n+1)\cdot{n!}  }\cdot\frac{n!}{cos(\pi{n}})\bigg|<1

    then by simplification we would have \lim_{n \to {\infty}}\bigg|\frac{cos(\pi(n+1))}{(n+1)cos(\pi{n  })}\bigg|<1...now how are you going to simplify that? this is why the direct comparison test is a better choice...hope this helps!
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