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Math Help - Series (ratio and root test)

  1. #1
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    Series (ratio and root test)

    (n^sqrt(2) ) / 2^n

    I want to determine if this series will converge or diverge. When I use the root test I get sqrt(2)/2 which is less than 1 so it will converge. The book used the ratio test. Could someone please show me how they used the ratio test to say it converges. I got lim n->infinity n/2 which is infinity so it would diverge.
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  2. #2
    Super Member Deadstar's Avatar
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    Oct 2007
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    You want to see if  \lim_{n \to \infty} \mid \frac{a_{n+1}}{a_n} \mid < 1.

    For your series you have...

    \frac{(n+1)^{\sqrt{2}}}{2^{n+1}} \cdot \frac{2^{n}}{n^{\sqrt{2}}}

    = \left ( \frac{n+1}{n} \right )^{\sqrt{2}} \cdot \frac{1}{2} \to \frac{1}{2} as n \to \infty
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