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Homework Help: Converge absolutely or conditionally, or diverges?

  1. Jan 30, 2007 #1
    1. The problem statement, all variables and given/known data

    Determine if the series converges absolutely, converges conditionally, or diverges.

    equation is here: http://img409.imageshack.us/img409/7353/untitledly5.jpg [Broken]

    2. Relevant equations

    maybe alternating series, or harmonic series?

    3. The attempt at a solution

    not real familiar with tan with series.
    haven't tried much, need supporting work for the answer.
    need help.
     
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jan 31, 2007 #2
    As n-> infinity, tan(1/n) -> tan(0) -> 0
    Does this help?
     
  4. Jan 31, 2007 #3

    quasar987

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    Actually, this says nothing at all about the series. The implication is one way only: "Sum a_n converges ==> a_n-->0" but "a_n-->0 ==> nothing".

    Actually the series satisfies all the criteria corresponding to the convergence of an alternating series. Remains to see if it converges absolutely. I.e. does

    [tex]\sum_{n=1}^{\infty}\tan(n^{-1})<\infty[/tex]

    ??
     
  5. Jan 31, 2007 #4
    no it doesn't converge absolutely because it continues on to infinity.

    however, i do ask, how do you know to test it to be less than infinity? in other words, the convergence for a alternating series passes. but what other series convergence did not pass?

    so ultimately, this will converge conditionally.

    for my work, i could prove this by showing the alternating series? and then showing that it also continues on to infinity?

    thanks again for all the help so far.
     
  6. Jan 31, 2007 #5

    quasar987

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    What do you mean by "continues on to infinity" ?
     
  7. Jan 31, 2007 #6

    mjsd

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    rcmango, i think you mean using the Leibniz test (for alternating series)
    there are three conditions, check all to prove.
     
  8. Jan 31, 2007 #7

    HallsofIvy

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    It is more to the point that tan(1/n) is decreasing. Knowing that it goes to 0 is neither necessary nor helpful.
     
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