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Divergence of a particular serie

  1. Jan 24, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]\sum_{n=1}^{\infty}{(-1)^n*n}[/itex]

    3. The attempt at a solution

    Well, this is a series I came upon when analyzing the endpoints of a particular power series, the thing is, my book says it's divergent by the Test for divergence, however, I can't find this result, what I tried to do so far:

    Alternating testing series: Of course, my first thought, the test fails because [itex]b_{n+1} > b_n[/itex], and not the opposite.

    Then I tried the test for divergence:
    [itex]\lim_{n->\infty}{(-1)^n*n}[/itex]
    But I can't solve this limit, I could solve it without the [itex](-1)^n[/itex]. But I can't just "take out the term I don't like and solve it".

    What am I doing wrong?

    Thanks in advance.
     
  2. jcsd
  3. Jan 24, 2013 #2

    Mark44

    Staff: Mentor

    Some books call this the "Nth term test for divergence." This theorem is very simple to use, and should probably be used first. It says that in a series ##\sum a_n##, if ##\lim_{n \to \infty} a_n ≠ 0##, then the series diverges.

    Note that this theorem can be used only on series for which ##\lim_{n \to \infty} a_n ≠ 0##. If the limit equals zero, the series could converge or it could diverge.
    This test doesn't fail - it is not applicable here.
    Expand the series by writing a few of the terms. It should be obvious what this series is doing.
     
    Last edited: Jan 25, 2013
  4. Jan 24, 2013 #3
    Yes indeed, I can see that it obviously diverge if I expand the series, I even tried graphing a few terms to look at that, but what I meant was, is there a "formal way" to write that?
     
  5. Jan 24, 2013 #4
  6. Jan 24, 2013 #5

    Mark44

    Staff: Mentor

    If the limit does not exist, it is automatically not zero, so it isn't necessary to include the part about it not existing. Perhaps it is added above for clarity.
     
  7. Jan 25, 2013 #6
    You would write something like "since the limit doesn't exist, the series diverges by the divergence test".
     
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