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Math Help - Determining Convergence or Divergence

  1. #1
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    Determining Convergence or Divergence

    Hello,

    I'm having difficulty determining the behavior of this series.

    "Show that the series: \sum_{n=1}^{\infty} \frac{5n+1}{8n^2-10n-52} converges or diverges by finding a p-series, \sum_{n=1}^{\infty}\frac{c}{n^p} or geometric series, \sum_{n=1}^{\infty}cr^n to use in the comparison test.

    Your answer should consist of four items separated by commas.
    1. The first item should be either the word less or the word greater.
    2. The second item should be an expression for the term bn of the series used in the comparison test.
    3. The third item should be a natural number N so that the comparison inequality an < bn holds for all n > N or an > bn holds for all n > N
    4. The fourth item should be the word converges or the word diverges to indicate whether you are saying that the given series Sum(an,n=1..infinity, i.e., , converges or diverges."
    I compared \frac{5n+1}{8n^2-10n-52} with \frac{5}{8n}

    saying that

    \frac{5n+1}{8n^2-10n-52} > \frac{5}{8n} for n > 4.

    And since \frac{5}{8n} is divergent by p-series, that

    \frac{5n+1}{8n^2-10n-52} was also divergent.

    So my answer looks like:

    greater,(5/8)(1/n),4,diverges

    which is incorrect. If anyone can point out where I've gone wrong, I would greatly appreciate it.
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  2. #2
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    Quote Originally Posted by auslmar View Post
    Hello,

    I'm having difficulty determining the behavior of this series.

    "Show that the series: \sum_{n=1}^{\infty} \frac{5n+1}{8n^2-10n-52} converges or diverges by finding a p-series, \sum_{n=1}^{\infty}\frac{c}{n^p} or geometric series, \sum_{n=1}^{\infty}cr^n to use in the comparison test.

    Your answer should consist of four items separated by commas.
    1. The first item should be either the word less or the word greater.
    2. The second item should be an expression for the term bn of the series used in the comparison test.
    3. The third item should be a natural number N so that the comparison inequality an < bn holds for all n > N or an > bn holds for all n > N
    4. The fourth item should be the word converges or the word diverges to indicate whether you are saying that the given series Sum(an,n=1..infinity, i.e., , converges or diverges."
    I compared \frac{5n+1}{8n^2-10n-52} with \frac{5}{8n}

    saying that

    \frac{5n+1}{8n^2-10n-52} > \frac{5}{8n} for n > 4.

    And since \frac{5}{8n} is divergent by p-series, that

    \frac{5n+1}{8n^2-10n-52} was also divergent.

    So my answer looks like:

    greater,(5/8)(1/n),4,diverges

    which is incorrect. If anyone can point out where I've gone wrong, I would greatly appreciate it.

    Well that looks OK to me, so perhaps its just being over restrictive in what it will accept, try:

    greater,5/(8n),4,diverges

    or:

    greater,5/(8*n),4,diverges

    or:

    greater,(5/8)(1/n),3,diverges

    (3 will do here since strict inequalities are involved, but the answer would be right for any value greater than or equal to 3)

    RonL
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  3. #3
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    Thanks for the help.

    Actually, it accepted:

    greater,(5)/(8*n),5,diverges
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  4. #4
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    Quote Originally Posted by auslmar View Post
    it accepted:
    greater,(5)/(8*n),5,diverges
    That is odd! That change is exactly what I had said in the post that I deleted.
    But when I ran it through MathCad, N>=4 worked.
    I wonder if the author of this problem made the same mistake that I did.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Plato View Post
    That is odd! That change is exactly what I had said in the post that I deleted.
    But when I ran it through MathCad, N>=4 worked.
    I wonder if the author of this problem made the same mistake that I did.
    There are two changes from the original submission I wonder if it needed both, or if just one of them would have worked?

    RonL
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