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Proving convergence of infinite series

  1. Oct 22, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\sum \frac{(-1)^{n}}{n+n^{2}}[/tex]

    Does this series converge as n -> infinity?

    2. Relevant equations

    3. The attempt at a solution

    First, by the absolute convergence test, [tex]\sum \frac{(-1)^{n}}{n+n^{2}}[/tex] should converge if [tex]\sum \left|\frac{(-1)^{n}}{n+n^{2}}\right|[/tex] converges.

    Second, [tex]\sum \left|\frac{(-1)^{n}}{n+n^{2}}\right| = \frac{1}{n+n^{2}}< \sum 1/n^{2}[/tex]

    Because the sum 1/n^2 converges, by the comparison test, [tex]\sum \left|\frac{(-1)^{n}}{n+n^{2}}\right|[/tex] converges.

    Which means that [tex]\sum \frac{(-1)^{n}}{n+n^{2}}[/tex] converges as well (by the absolute convergence test).
  2. jcsd
  3. Oct 22, 2009 #2
    Your proof appears to be valid.
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