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Math Help - Another Divergence/Convergence Test

  1. #1
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    Another Divergence/Convergence Test

    Hello,

    So, I've got yet again another problem where I'm not sure if I made an error or the system is fickle. The problem is as follows:

    "Show that the series: \sum_{n=0}^{\infty}\frac{(4^n-2n^6)}{(8(3^n)+5n^4)}

    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 limit comparison test. Your answer should consist of three items separated by commas.
    1. The first item should be an expression for the term b_n of the series used in the limit comparison test.
    2. The second item should be a non-zero number L so that the limit of \frac{a_n}{b_n} is L.
    3. The third item should be the word converges or the word diverges to indicate whether you are saying that the given series converges or diverges."
    So, I used \frac{4^n}{8(3^n)}

    and I have:

    L = \lim_{n \to \infty}\frac{\frac{4^n-2n^6}{8(3^n)+5n^4}}{\frac{4^n}{8(3^n)}} = \lim_{n\to\infty}\frac{(4^n-2n^6)(8(3^n))}{(8(3^n)+5n^4)(4^n)} = \lim_{n\to\infty}\frac{(3^n)(4^n)(8-16n^6(1/4^n))}{(3^n)(4^n)(8+5n^4(1/3^n))} = \lim_{n\to\infty}\frac{8-16n^6(1/4^n))}{8+5n^4(1/(3^n))} =  \frac{8}{8} = 1 \ne 0

    therefore, \sum_{n=0}^{\infty}\frac{(4^n-2n^6)}{(8(3^n)+5n^4)} converges.

    and my answer looks like:

    \frac{(4^n)}{8(3^n)},1,converges

    which is not recognized as correct by the system. So, I'm wondering if I've faulted somewhere along the lines or if the system is just being picky. If you can offer any hints or ideas, I would be grateful.

    Thanks

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  2. #2
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    Quote Originally Posted by auslmar View Post
    my answer looks like:

    \frac{(4^n)}{8(3^n)},1,converges

    which is not recognized as correct by the system. So, I'm wondering if I've faulted somewhere along the lines or if the system is just being picky. If you can offer any hints or ideas, I would be grateful.

    Thanks

    Hmm I am sorry I misread the your question initially. I think your first and second answers are right.

    However \frac{(4^n)}{8(3^n)} diverges since \frac43 > 1. Thus by limit comparison test the answer is diverges...
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  3. #3
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    Quote Originally Posted by Isomorphism View Post
    Its wrong because in ratio test, if L = 1 or if the limit does not exist, then the test is inconclusive. So you cant conclude it converges by that sequence...



    Oh yes, I forgot about this! Hmm yes, well. Something else is wrong, because the system doesn't allow me to answer "inconclusive" or anything similar. And, it neither accepts "converges" or "diverges" in conjunction with my previous answers.
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  4. #4
    Lord of certain Rings
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    Quote Originally Posted by auslmar View Post
    Oh yes, I forgot about this! Hmm yes, well. Something else is wrong, because the system doesn't allow me to answer "inconclusive" or anything similar. And, it neither accepts "converges" or "diverges" in conjunction with my previous answers.
    Its not wrong actually... but lets fiddle a little...

    Try
    \left(\frac{4}{3}\right)^n,\frac18,diverges
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  5. #5
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    Quote Originally Posted by Isomorphism View Post
    Its not wrong actually... but lets fiddle a little...Try
    \left(\frac{4}{3}\right)^n,\frac18,diverges
    Well, I'll be. The system took that one! Thanks for the help.
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