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Series and Limits Problem

  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data
    I am supposed to determine whether the summation attached is convergent or divergent

    2. Relevant equations
    Alternating Series Test
    Test for Divergence

    3. The attempt at a solution
    The attempted solution is attached. Using the two different tests I am getting two different answers.
     

    Attached Files:

  2. jcsd
  3. Mar 29, 2016 #2

    LCKurtz

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    It is much preferred for you to type the problems rather than post a download.

    You have ##\frac 1 {\sqrt{n+1}}\to 0## which is correct. Now since$$
    0 \le \left | \frac {(-1)^n} {\sqrt{n+1}}\right | \le \frac 1 {\sqrt{n+1}}$$ how could the alternating one not go to zero? And, by the way, ##(-1)^\infty## makes no sense.
     
    Last edited: Mar 29, 2016
  4. Mar 29, 2016 #3
    Okay, you used the squeeze theorem which makes sense, but why doesn't the test for divergence work? Isn't (-1) undefined meaning the limit is undefined meaning the series is divergent?
     
  5. Mar 29, 2016 #4

    LCKurtz

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    Yes, as I said, ##(-1)^\infty## makes no sense or, as you say, is undefined. What is happening in this problem is that the denominator is getting larger and the numerator is either plus or minus 1 for any n. The fraction gets small no matter the sign, so regardless of the alternating sign the fraction goes to zero.
     
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