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Calculus II - Series and Convergence

  1. Aug 18, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine if the series

    inf
    Sigma n/(2n+1)
    n=1

    converges

    2. Relevant equations



    3. The attempt at a solution

    When i did this I originally I thought I would just apply the divergence test

    lim n/(2n+1) =/= 0
    n->inf

    there fore I thought by the divergence test the series diverges but I guess sense the limit is defined to be 1/2 it converges...

    I'm confused... I think I may be over complicating this but by the divergence test shouldn't this series diverge sense the limit does not equal zero?

    Thanks for any help
     
  2. jcsd
  3. Aug 18, 2011 #2
    Yes, that's correct. It's more commonly called the nth term test for divergence.
     
  4. Aug 18, 2011 #3

    gb7nash

    User Avatar
    Homework Helper

    The bolded part is correct. Since you took the limit of the summand and received 1/2, the limit diverges. If the sum converges its limit is equal to 0.

    However, the converse of the statement is not true. Don't fall into the trap of thinking the sum converges if the limit is equal to 0. If you obtain a limit of 0, you need to resort to a different method to determine if the sum converges.
     
  5. Aug 18, 2011 #4
    hm thanks i think i got it mixed up and it asked me if the sequence converges which it does lol
     
  6. Aug 18, 2011 #5
    No it doesn't. It diverges. You just showed this using your test. Or was that a typo?

    And it's a series, not a sequence.
     
  7. Aug 18, 2011 #6
    ya i got it and understand now i got a different part mixed up and stuff
     
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