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Math Help - Series/convergence problem

  1. #1
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    Series/convergence problem

    Summation notation from n=1 to n=infinity

    sin(n)/n

    I believe it converges by the ratio test but my book says it's "inconclusive" because of a comparison to the harmonic series. I don't get it....... any help please?
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  2. #2
    Member Nacho's Avatar
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    <br />
\sum\limits_{n \geqslant 1} {\sin \left( n \right)}  < K_{ \in \mathbb{R}} {\text{ }}<br />
(you try prove it) and <br />
\frac{1}<br />
{n}\xrightarrow[{n \to \infty }]{}0<br />
for dirichlet, the serie <br />
\sum\limits_{n \geqslant 1} {\frac{{\sin \left( n \right)}}<br />
{n}} <br />
converge
    Last edited by Nacho; February 26th 2009 at 06:28 AM.
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  3. #3
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    \sum\limits_{n=1}^{\infty }{a_{n}\sin n}<\infty whenever a_n is a decreasing sequence and \lim_{n\to\infty}a_n=0.

    Here a_n=\frac1n and this fulfills the above conditions, whereat \sum\limits_{n=1}^{\infty }{\frac{\sin n}{n}}<\infty , and we're done.
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  4. #4
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    Quote Originally Posted by Kaitosan View Post
    Summation notation from n=1 to n=infinity

    sin(n)/n

    I believe it converges by the ratio test but my book says it's "inconclusive" because of a comparison to the harmonic series. I don't get it....... any help please?
    That's very strange. You can say that a particular test for convergence is "inconclusive" but that doesn't apply to a series itself! Obviously, any given series either converges or it doesn't. That's true whether we know which is true or not!
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