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Homework Help: Prove that a series converges.

  1. Dec 1, 2012 #1
    1. The problem statement, all variables and given/known data
    $$ \sum\limits_{n=1}^\infty \frac{1 - (-1)^n e}{1 + (n \pi)^2}$$
    2. Relevant equations

    3. The attempt at a solution
    I'd imagine you have to use the comparison test on it but am unable to figure out where to start. Any suggestions would be greatly appreciated.
    Last edited: Dec 1, 2012
  2. jcsd
  3. Dec 1, 2012 #2


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    Gold Member

    Well indeed the comparison test will do (with 1/n^2).

    I'll let you see how exactly.
  4. Dec 1, 2012 #3


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    Homework Helper

    Interesting sum, compare to 1/n^2
  5. Dec 1, 2012 #4
    Last edited: Dec 1, 2012
  6. Dec 1, 2012 #5


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    Gold Member

    Much simpler than that:

    [tex]|\frac{1-(-1)^ne}{1+(\pi n)^2}| \leq \frac{1+e}{1+(\pi n)^2}[/tex]
  7. Dec 1, 2012 #6
    oh so could i just have done:

    [tex]0 \leq |\frac{1-(-1)^ne}{1+(\pi n)^2}| \leq |\frac{1}{n^2}|[/tex]
    [tex]\therefore \frac{1-(-1)^ne}{1+(\pi n)^2}[/tex] absolutely converges.
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