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Infinite Series

  1. Nov 19, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that the series converges. Then compute an estimate of the limit that is guaranteed to be in error by no more than 0.005

    [tex]\sum_{k=5}^{\infty} (-1)^k \frac{k^{10}}{10^k}[/tex]

    3. The attempt at a solution

    This is obviously an alternating series and I know that

    [tex]C_{k} = \frac{k^{10}}{10^k}[/tex]

    and I know that Cn+1 is greater than the absolute value of S-Sn. So I can set up to be something like

    [tex]\frac{(n+1)^{10}}{10^{n+1}} <= 0.005[/tex]

    the part I can't figure out is how to solve that for n. Is there a way to simplify that fraction? How would I solve this for n? Thanks

    Oh and I also couldn't figure out how to exactly show that the limit as k-> infinity of Ck goes to 0 without taking the derivative 10 times. So the whole problem I am having is with the fraction.
  2. jcsd
  3. Nov 19, 2008 #2


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    Homework Helper

    If you want to show the limit goes to zero, take the log and try showing that the limit of that goes to -infinity. Once you done that also find the value of x where x^10/10^x is a maximum by maximizing the log. Then you know it's decreasing after that value. No, I don't think you can really 'solve for n'. But once you know where the series is decreasing, just find an n so the term is less that 0.005.
  4. Nov 20, 2008 #3
    Thanks, that makes sense.
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