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Homework Help: Sequences and Series

  1. Jul 4, 2005 #1
    I need to determine whether Sigma [sin(1/x)] for x=1 to x=infinity converges or diverges. I have a feeling that it diverges, but I don't know how to prove it.

    Thanks
     
  2. jcsd
  3. Jul 4, 2005 #2

    Pyrrhus

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    Use the limit comparison test, with the harmonic series [itex] \sum_{n=1}^{\infty} \frac{1}{n} [/itex]
     
  4. Jul 4, 2005 #3
    Thanks Cyclovenom, I took [tex]\frac{sin(1/x)}{1/x}[/tex] and did the limit as x approaches infinity with L Hopital's rule, but I got 0, so doesn't that make it inconclusive?

    Thanks
     
  5. Jul 4, 2005 #4

    Pyrrhus

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    Your result throught L'Hospital is wrong, it will be 1, and sice 1 > 0, and this harmonic series diverges, then sin (1/x) diverges.

    Alternatively you could consider [itex] \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 [/itex]
     
  6. Jul 4, 2005 #5
    hello there

    just use the integral test

    [tex]\sum_{n=1}^{\infty} \sin{\frac{1}{n}} \le \int_{1}^{\infty}\sin{\frac{1}{x}}dx \not< \infty [/tex]

    its not finite and so does not converge

    steven
     
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