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Alternating Series Question

  1. Apr 20, 2013 #1
    1. The problem statement, all variables and given/known data

    The problem contained five answer choices, of which I the answerer was to find one that fit the criteria of the question. The question was: "Which series of the following terms would be convergent?".

    It listed five series, The answer was this term: 1 + (-1)n / n.


    2. Relevant equations

    [itex]\Sigma[/itex]1 + (-1)n / n.

    3. The attempt at a solution

    I find this very confusing simply because whilst separating the series into two separate series, [itex]\Sigma[/itex]1 and [itex]\Sigma[/itex](-1)n / n, The second series converges (yes I was surprised also) by the alternating series test. Originally, I was dumbfounded because of the absolute value test, so I suppose the series is conditionally convergent. Anyways, if [itex]\Sigma[/itex](-1)n / n is convergent and [itex]\Sigma[/itex]1 is divergent, 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.... + 1 .... +1 (you get the point) Then how can a finite number (the second alternating series) affect convergence? Also, the limit test kind of rules convergence out for this one. Ha.

    I may be missing something deceptively simple, so if anyone can help me out here that'd be great!
     
  2. jcsd
  3. Apr 20, 2013 #2

    jbunniii

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    Do you mean this series:
    $$\sum_{n=1}^{\infty}\left(1 + \frac{(-1)^n}{n}\right)$$
    The first question to ask is whether the terms converge to zero. If not, the series cannot possibly converge.
     
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