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Homework Help: Test the following Series for Convergence (absolute or conditional) or divergence

  1. Sep 15, 2010 #1
    Test the following Series for Convergence (absolute or conditional) or divergence

    ∑ (-1)^(n)/ (1+1/n)^(n^2)

    I know we solve it with the root test but i reached at a point where i don't know how to cancel it out

    ----

    lim(n--> infinity)= 1/(1+1/n+1)^(n+1^2) *[ (1+1/n)^(n^2)]/1

    What do i do after this?
     
  2. jcsd
  3. Sep 15, 2010 #2
    [tex]
    \ldots=\frac{(1+\frac{1}{n})^{n^2}}{(1+\frac{1}{n+1})^{n^2+2n+1}}=
    (1+\frac{1}{n+1})^{-2n-1}\left[\frac{(n+1)^2}{n(n+1)}\right]^{n^2}
    \sim e^{-2}\left(\left[1+\frac{1}{n(n+1)}\right]^{n^2+n}\left)^\frac{n^2}{n^2+n}
    \sim e^{-1}
    [/tex]
     
  4. Sep 15, 2010 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You are using the ratio test, not the root test. The problem is a lot easier if you actually use the root test.
     
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