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Alternating Series Question

  1. May 2, 2012 #1
    Prove that [tex] \sum^{∞}_{n=1}(-1)^{n} [/tex] diverges.

    I realized that the alternating series test can only be used for convergence and not necessarily for divergence. I might have to apply a ε-δ proof (Yikes!) which I have never been good at so please help me out.

  2. jcsd
  3. May 2, 2012 #2
    What about the sequence of terms?
  4. May 2, 2012 #3
    Hmm good point. If we can prove the sequence does not converge to 0, we have proved that the series diverges. How can we do that? Shall I look at the limit definition and look for an ε that invokes a necessary contradiction?

  5. May 2, 2012 #4
    Yep, looks like a good plan. Try to do that.
  6. May 2, 2012 #5


    Staff: Mentor

    Look at the Nth term test for divergence.
  7. May 2, 2012 #6


    User Avatar
    Science Advisor

    Seems to me the simplest thing to do is to show that the sequence of "partial sums",
    [itex]S_n= \sum_{i= 1}^n (-1)^n[/itex]
    does not converge.
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