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Series, is this allowed?

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Well I have a series that I solved one way, but my professor solved another and I'm wondering if my way is ok.
    [tex]\sum\limits_{n=1}^\infty \frac{(-5)^{2n}}{n^{2}9^{n}}[/tex]

    2. Relevant equations



    3. The attempt at a solution
    Alright well I started out by changing it to:
    [tex]\sum\limits_{n=1}^\infty \frac{1}{n^{2}}(\frac{(-5)^{2n}}{9^{n}})=\sum\limits_{n=1}^\infty \frac{1}{n^{2}}(\frac{25}{9})^{n}[/tex]

    So I concluded since the second part is a geometric series with r>1, it's divergent.

    Is this allowed?
     
  2. jcsd
  3. Apr 3, 2013 #2
    Your manipulations are legal, but how do you conclude that just because the "second part" of the series is divergent, that the entire series is divergent. There is, after all, a factor of [itex] \frac{1}{n^2} [/itex] that reduces each term. You could look at long term behavior of the terms however and note that [itex] \displaystyle\lim_{n\rightarrow \infty} {\frac{(\frac{25}{9})^n}{n^2}} = \infty [/itex].
     
  4. Apr 3, 2013 #3
    Very interesting, never thought about doing that. Could of just done the standard ratio test, which I think would work out easier, but hey this is a good way to do this problem I think.

    Thanks.
     
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