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Value of X in a series

  1. Oct 29, 2008 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution

    I'm testing the end points of 2 < x < 10.
    The Series looks like this:

    SUM[tex]\frac{(x-6)^n}{4^n\sqrt{n}}[/tex]

    The first endpoint (10), I got diverges by the p-test (p <= 1). The second one, in my notes it says to take the lim, and if it's 0 then it converges by alt. series test. Why do I do two different tests?
     
  2. jcsd
  3. Oct 29, 2008 #2

    Dick

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    You take two different tests because they are two different series. One converges (conditionally) and one doesn't. I don't see what the problem is.
     
  4. Oct 29, 2008 #3

    HallsofIvy

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    At x= 10, You have
    [tex]\sum \frac{4^n}{4^n\sqrt{n}}= \sum\frac{1}{n^{1/2}}[/tex]
    so the p test applies.

    At x=1 you have
    [tex]\sum \frac{(-5)^n}{4^n \sqrt{n}}= \sum \left(\frac{-5}{4}\right)^n\frac{1}{n^{1/2}}[/itex]
    which is an alternating series.
     
  5. Oct 29, 2008 #4
    Ok, maybe I just need help with my exponents. Why did the n exponents cancel in the first one but not in the second one?

    I got
    [tex]\sum \frac{(-4)^n}{4^n \sqrt{n}}= \sum \left(\frac{-4}{4}\right)^n\frac{1}{n^{1/2}} = \frac{-1}{n^{1/2}}[/tex]
    for the second one.
     
  6. Oct 29, 2008 #5

    HallsofIvy

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    Because 4/4= 1 and -5/4 does not!
     
  7. Oct 29, 2008 #6
    I edited my post. sorry
     
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