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Convergence of a Series

  1. May 7, 2009 #1

    fishingspree2

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    [tex]\sum_{k=1}^{\infty}\frac{2+\left(-1 \right)^{k}}{5^{k}}[/tex]

    Hello, I am trying to determine if this series converges or diverges. I have tried comparison test and d'Alembert's test but I was not successful

    Can anyone suggest me a test learned in Calculus 2 that will work?

    Thank you
     
    fishingspree2, May 7, 2009
  2. jcsd
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  3. May 7, 2009 #2

    Billy Bob

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    Break it into a sum of two series?
     
    Billy Bob, May 7, 2009
  4. May 7, 2009 #3

    Cyosis

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    Homework Helper

    Since you're asking if there exists another test you could also use.

    [tex]
    \lim_{n \to \infty}\sqrt[n]{|a_n|}<1
    [/tex]

    It converges absolutely if the limit is smaller than 1.
     
    Cyosis, May 7, 2009
  5. May 7, 2009 #4

    fishingspree2

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    Thank you, I have figured it out. I have another question if you don't mind

    [tex]\sum_{k=1}^{\infty}\frac{\left|\cos k \right|}{k^{3}}[/tex]

    It looks like a riemann series but I don't really know how to deal with trigonometric functions in series yet. What to do? Can anyone explain me?

    Thank you
     
    fishingspree2, May 7, 2009
  6. May 7, 2009 #5

    VeeEight

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    What values can |cos k| take? Perhaps consider a p-series
     
    VeeEight, May 7, 2009
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