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Another series problem

  1. Jan 24, 2007 #1
    1. The problem statement, all variables and given/known data

    Does this problem converge absolutely, conditionally, or does it diverge?

    the equation: [URL [Broken][/URL]

    2. Relevant equations

    also, the hint is to first show that ln(1 + x) <= x if x > 0

    3. The attempt at a solution

    It looks like an alternating series. not sure what the hint is implying or if its converging.

    Thanks for any help.
     
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jan 24, 2007 #2

    Gib Z

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    Homework Helper

    Each term gets smaller and smaller, and converges to zero. It is absolutely convergent.

    The ratio test will tell you it converges as well.
     
    Last edited: Jan 24, 2007
  4. Jan 24, 2007 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Is that what you mean to say? Each term of
    [tex]\Sigma_{n\rightarrow \infty}\frac{1}{n}[/itex]
    "gets smaller and smaller, and converges to zero" but the series doesn't converge at all.
     
  5. Jan 24, 2007 #4

    mjsd

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    Homework Helper

    when it is an alternating series you can use Leibniz test
    your pic is not very clear... but my guess is that the hint is to help you establish one of the condition in the Leibniz test namely, the terms are getting smaller

    Leibniz test:
    If [tex]\sum_1^{\infty} (-1)^{n+1} b_n[/tex] such that all [tex]b_n>0[/tex] (ie alternating series) and [tex]b_{n+1} < b_n\; \forall\,n[/tex] and [tex]b_n\rightarrow 0[/tex], then series converges to S and [tex]|S-S_k|\leq b_{k+1}[/tex]
     
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