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Radius of Divergence

  1. Nov 14, 2014 #1
    1. The problem statement, all variables and given/known data
    I have this problem to consider the power series,
    [itex]\sum_{n=1}^{\infty}\frac{(-4)^{n}}{\sqrt{n}}(x+4)^{n}[/itex]
    So, i need to find the [itex]R[/itex] and interval of convergence.
    2. Relevant equations

    3. The attempt at a solution

    This is what i did:
    [itex]\lim_{n\rightarrow \infty} {\frac{(-4)^{n+1}(x+4)^{n+1}}{\sqrt{n+1}}}\frac{\sqrt{n}}{(-4)^{n}(x+4)^{n}}[/itex]

    and this is what i get after i finished calculating for [itex] R [/itex] [itex]= 4|x+4|[/itex] [itex]\rightarrow R = 1/4[/itex] and the interverval for convergence [itex]= (-17/4, -15/4)[/itex]

    When i submitted this answer into webwork, the system said it was wrong. So, can somebody please guide me to the correct path of calculating this question please.
     
  2. jcsd
  3. Nov 14, 2014 #2

    Zondrina

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    Homework Helper

    Your answer looks correct to me.
     
  4. Nov 14, 2014 #3
    Yes, the answer is correct but i just noticed that i enter the interval in the wrong notations. It supposed to be [itex](-17/4,-15/4][/itex]. I just don't understand why one interval is open, and the other one is closed.
     
  5. Nov 14, 2014 #4
    When you use the Ratio test for interval of convergence, you have to check the end points, this is because the test is inconclusive when the limit is 1.
     
  6. Nov 14, 2014 #5

    mfb

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    2016 Award

    Staff: Mentor

    You have to check the borders separately. If you do that, you'll see one gives a convergent series, the other one does not.

    Edit: Didn't see Panphobia's post before.
     
  7. Nov 14, 2014 #6
    So, when one point is convergent we use the closed interval, and open if it diverges?
     
  8. Nov 14, 2014 #7
    Yes, because it is included in the interval.
     
  9. Nov 14, 2014 #8
    ok that makes sense. Thanks for the replies and the help.
     
  10. Nov 15, 2014 #9

    Mark44

    Staff: Mentor

    Technical point. (-17/4, -15/4] is one interval. The two numbers are endpoints of this interval, not intervals themselves.
     
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