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Homework Help: Absolute Convergence

  1. Apr 4, 2013 #1
    1. The problem statement, all variables and given/known data
    Absolute, Conditional, - convergence, or Divergence.

    2. Relevant equations
    [itex]\displaystyle \sum^{∞}_{n=1} (-1)^n e^{-n} [/itex]

    3. The attempt at a solution
    1. Alternating Series Test
    2. Ratio Test for ABsolute Convergence

    1. [itex]\displaystyle (-1)^n (1/e)^n [/itex]
    an > 0 for n=1,2,3,4 - YES
    decreasing - YES
    limit (n->inf) = 0 - YES


    2.[itex]\displaystyle (1/e)^{n+1} * (e^n) = 1/e[/itex]
    limit (n->inf) = 1/e

    1/e < 1

    Thus, Absolute Convergence.
  2. jcsd
  3. Apr 4, 2013 #2


    Staff: Mentor

    Yes, the series converges, and converges absolutely. Your work is a little sketchy in parts, so if your instructor is picky, you might lose points. For example, in the alt. series test you need to show that an > 0 for all n ≥ 1, not just n = 1, 2, 3, and 4. And you don't show any work that justifies your saying that the sequence an is decreasing or that the limit of the sequence is zero.
    Last edited: Apr 4, 2013
  4. Apr 4, 2013 #3
    i did not show the work here, but i did show the work for decreasing by setting f(x) = series and taking derivative.
    for the limit i have lim(x->inf) (1/e)^n = 0 (not sure how to show that this is zero)?

    also, how do i show that an > 0 for all n >= 1 ?
  5. Apr 4, 2013 #4


    Staff: Mentor

    The limit as you wrote it doesn't make much sense, since you have x increasing, but the expression you're taking the limit of doesn't involve x.

    One way to show that this limit is zero is to show that (1/e)n can be made arbitrarily close to zero. IOW, using the definition of the limit, which in this case involves N and ##\epsilon##. I don't know if actual limit definitions have been presented in your class just yet.
    Here, an = 1/en. The only way for a rational expression to be equal to zero is for its numerator to equal zero. Obviously that can't happen here. The denominators in this sequence, e1, e2, ..., en, ... are an increasing sequence of positive numbers, so it must be true that their reciprocals are decreasing and positive. An induction proof, which would be easy, would be convincing here.
  6. Apr 5, 2013 #5
    everytime we take a limit, i don't think the instructor wants us to use the definition of the proof. also, it's been a bit since i took a number theory class (discrete mathematics) which involved induction proofs.

    i stated what you did above about the denominators and their reciprocals, but i have no other proof than that. also, i'm not sure what i would say for my other problems.
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