Sequences and Series

  • Thread starter Phoenix314
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  • #1
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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
 

Answers and Replies

  • #2
Pyrrhus
Homework Helper
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Use the limit comparison test, with the harmonic series [itex] \sum_{n=1}^{\infty} \frac{1}{n} [/itex]
 
  • #3
16
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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
 
  • #4
Pyrrhus
Homework Helper
2,178
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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]
 
  • #5
176
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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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