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How to prove Convergence of this Series

  1. Feb 28, 2016 #1
    1. The problem statement, all variables and given/known data

    Use any appropriate test to determine the convergence or divergence of the following series:

    [tex] \sum_{i=0}^{\infty} \frac{2^{i} + 3^{i}}{4^{i}+5^{i}} [/tex]

    2. Relevant equations

    3. The attempt at a solution

    I've run it through mathematica and it told me it's convergent. However, I can't seem to find the right test to use. Ratio test / Limit comparison doesn't seem to work as nothing cancels, I can't find a test series for direct comparison, and root test wouldn't work? Have I over looked anything?
  2. jcsd
  3. Feb 28, 2016 #2


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    Gold Member

    You can start with an asymptotic approach ##\sum_{i=0}^{\infty}\frac{2^{i}+3^{i}}{4^{i}+5^{i}}\sim \sum_{i=0}^{\infty}\frac{3^{i}}{5^{i}}##
  4. Feb 28, 2016 #3


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    Science Advisor
    Homework Helper

    Try comparison test.
    Hint: if 0<a<b and 0<c, 0<d, then ##\frac{a+b}{c+d}\leq \frac{2b}{d}##
  5. Feb 28, 2016 #4


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    Gold Member

    How about : ## 0<a<b<c<d ## then ##\frac {a+b}{c+d} ##?
  6. Feb 28, 2016 #5
    Alright I think I got it now, thanks.
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