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Math Help - What Test Do I Use and How?

  1. #1
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    What Test Do I Use and How?

    Could someone explain how to use what tests to determine whether the sum from n=1 to infinity of (cos(n+9)/(n^(3/2))) is absolutely convergent, conditionally convergent, or divergent? Could I use the direct comparison or limit comparison for this? If so, how; what would I compare it to?
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    Re: What Test Do I Use and How?

    Since \displaystyle \left| \cos{(x)} \right| \leq 1 for all x, that means your absolute value series will be less than the series of \displaystyle \frac{1}{n^{\frac{3}{2}}} .

    This is a convergent p-series, so by the comparison test, your series is absolutely convergent.
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    Re: What Test Do I Use and How?

    Quote Originally Posted by Prove It View Post
    Since \displaystyle \left| \cos{(x)} \right| \leq 1 for all x, that means your absolute value series will be less than the series of \displaystyle \frac{1}{n^{\frac{3}{2}}} .

    This is a convergent p-series, so by the comparison test, your series is absolutely convergent.
    So, if I were to use the limit comparison test, would I come to the same conclusion? Is it true that if b(sub n) converges and the limit as n goes to infinity of (a(sub n)/b(sub n)) approaches infinity, then a(sub n) also converges?
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    Re: What Test Do I Use and How?

    That's exactly what I did use...
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    Re: What Test Do I Use and How?

    Quote Originally Posted by Preston019 View Post
    So, if I were to use the limit comparison test, would I come to the same conclusion? Is it true that if b(sub n) converges and the limit as n goes to infinity of (a(sub n)/b(sub n)) approaches infinity, then a(sub n) also converges?
    There is no need for any further tests.
    \sum\limits_{n = 1}^\infty  {\left| {\frac{{\cos (n)}}{{\sqrt {{n^3}} }}} \right|}  \le \sum\limits_{n = 1}^\infty  {\frac{1}{{\sqrt {{n^3}} }}} tells you that your series converges absolutely.

    If a series converges absolutely it converges period.
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    Re: What Test Do I Use and How?

    Quote Originally Posted by Prove It View Post
    That's exactly what I did use...
    You just said comparison test, you didn't specify whether you used the direct comparison test or the limit comparison test.

    Would my second question be true then?
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    Re: What Test Do I Use and How?

    Quote Originally Posted by Plato View Post
    There is no need for any further tests.
    \sum\limits_{n = 1}^\infty  {\left| {\frac{{\cos (n)}}{{\sqrt {{n^3}} }}} \right|}  \le \sum\limits_{n = 1}^\infty  {\frac{1}{{\sqrt {{n^3}} }}} tells you that your series converges absolutely.

    If a series converges absolutely it converges period.
    I know, I'm just trying to figure out whether
    ...if b(sub n) converges and the limit as n goes to infinity of (a(sub n)/b(sub n)) approaches infinity, then a(sub n) also converges?
    is true.
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