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Homework Help: Real Analysis, Series

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data
    a) Show that the series ∑ from n = 1 to infinity 1/n^p where p converges when p > 1 and
    diverges for p=1.

    b) Prove that the following series diverges: ∑ from n = 1 to infinity sqrt(n)/n+1

    c) Use an appropriate test to show whether ∑ from n = 1 to infinity [(−1)^n * n^2/(n^2 +1)] converges or diverges.

    d) For what values of x , if any, does the following series converge? Show
    how you arrived at your answer.
    ∑ from n = 1 to infinity [(x^(2n + 1))/(2n + 1)!]

    3. The attempt at a solution
    Im lost completely
  2. jcsd
  3. Apr 22, 2012 #2


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    Science Advisor
    Homework Helper
    Gold Member

    I'm sorry to hear that. If you are that lost now at the end of the semester, I'm guessing you will be repeating your course. You aren't going to find anyone here to work those for you without you showing some work of your own.
  4. Apr 22, 2012 #3
    http://dl.dropbox.com/u/33103477/222.jpg [Broken]
    http://dl.dropbox.com/u/33103477/222222.jpg [Broken]

    b)Use a comparison test, (n+1)>(n+1)/sqrtn so 1/(n+1)<sqrt(n)/(n+1).
    Now 1/(n+1) diverges so the series you want also diverges.

    c) Use the fact that an absolutely convergent series converges, so just prove, n^2/n^2+1 convereges to 1 using limits hence the whole damn thing converges.

    d)try a ratio test
    Last edited by a moderator: May 5, 2017
  5. Apr 22, 2012 #4
    @sid Thank you a bunch!
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