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Math Help - Determine convergence or divergence

  1. #1
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    Determine convergence or divergence

    where ak= (7^n)/(3n^6+4^n)

    I tried the alternating series test but I get stuck on determining whether or not it is decreasing (using derivative)

    Also this other problem test for absolute convergence, conditionally convergent or divergent
    infinity
    E (-1)^n (4n^8+4)/(8n^9+2)
    n=1

    how would i test this for absolute convergence? I tried ratio test and it just becomes a bigger mess
    Last edited by krzyrice; May 17th 2010 at 07:16 PM.
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  2. #2
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    Quote Originally Posted by krzyrice View Post
    where ak= (7^n)/(3n^6+4^n)

    I tried the alternating series test but I get stuck on determining whether or not it is decreasing (using derivative)

    Also this other problem test for absolute convergence, conditionally convergent or divergent
    infinity
    E (-1)^n (4n^8+4)/(8n^9+2)
    n=1

    how would i test this for absolute convergence? I tried ratio test and it just becomes a bigger mess

    The first sum seems to be \sum^\infty_{k=1}(-1)^k\frac{7^k}{3k^6+4^k} (I assume here that you meant the index to be k and not n...), and if this is so the sum can't converge since

    the limit of its general term is not zero: \frac{7^k}{3k^6+4^k}\geq \frac{7^k}{2\cdot 4^k}\xrightarrow [k\to\infty]{}\infty , and with the alternating sign the general term doesn't even diverge to infinity.

    About the second one: it converges conditionally since it is a Leibnitz series, but taking its general term's absolute value we get:

    \frac{4n^8+4}{8n^9+2}\geq \frac{4n^8}{16n^9}=\frac{1}{4}\,\frac{1}{n} , and by the comparison test with the harmonic series it diverges.

    Tonio
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