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Calculus II - Alternating Series Test - Convergent?

  1. Apr 2, 2012 #1
    Hello! I was working some practice problems for a Calc II quiz for Friday on the alternating series test for convergence or divergence of a series. I ran into a problem when I was working the following series, trying to determine whether it was convergent or divergent:

    1. The problem statement, all variables and given/known data

    [itex]\sum[/itex] (-1)n[itex]\frac{3n-1}{2n+1}[/itex]
    n=1


    2. Relevant equations
    Alternating Series Test:
    For an alternating series


    [itex]\sum[/itex] (-1)n-1bn
    n=1

    to converge, it must satisfy the following conditions:

    1. bn+1 ≤ bn for all n in the series
    2. lim bn = 0
    n[itex]\rightarrow[/itex]∞

    Test for Divergence of a Series:
    If the limit as n approaches infinity of a series does not exist or does not equal 0, the series is divergent.



    3. The attempt at a solution
    I first began by analyzing the limit as n approaches infinity for the series. By dividing the coefficients of the highest exponential power of the top and bottom (in this case, 1), I found the limit of the series to be [itex]\frac{3}{2}[/itex]. It did not satisfy the second condition of the alternating series test, and as such, I sought to double-check with the Test for Divergence of a Series.

    (-1)n's limit does not exist, as it is always alternating back and forth between -1 and 1.

    [itex]\frac{3n-1}{2n+1}[/itex]'s limit, as mentioned above, I found to be [itex]\frac{3}{2}[/itex].

    From this information, I concluded that the limit did not exist, and that the series was divergent.

    I decided to check the answer from the back of the textbook, and it said it was convergent! I ran through it again a couple of times and got the same result: divergent. I'm not sure what I did wrong here - could I please have some insight from someone else to shed some light on the situation? Thank you very much!
     
    Last edited: Apr 2, 2012
  2. jcsd
  3. Apr 2, 2012 #2
    I'll tell you what's wrong here: your book. That series is definitely not convergent.
     
  4. Apr 2, 2012 #3
    Man, you've got to be kidding me. Nobody's perfect, I guess. I'll be sure to tell my professor in the morning. Thanks for the help!
     
  5. Apr 3, 2012 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The most fundamental theorem about infinite series, typically the first proved in a textbook, is
    If [itex]\lim_{n\to\infty} a_n\ne 0[/itex], then [itex]\sum a_n[/itex] does NOT converge.
     
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