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Math Help - Test the series for convergence

  1. #1
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    Test the series for convergence

    I am struggling through this new material.
    Here is the problem: Test the series for convergence-photo-5.jpg

    I know from the limit comparison test that both b_n and a_n either converge or diverge, but I cant seem to find out which of the two it is...

    How can I find whether b_n is convergent or divergent??

    Thank you for the help!
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  2. #2
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    first, note that ...

    \displaystyle \frac{1 + 2^n}{1 + 3^n} = \frac{1}{1+3^n} + \frac{2^n}{1+3^n}


    individually ...

    \displaystyle \frac{1}{1+3^n} < \frac{1}{3^n} = \left(\frac{1}{3}\right)^n

    \displaystyle \frac{2^n}{1+3^n} < \frac{2^n}{3^n} = \left(\frac{2}{3}\right)^n

    ... you should be able to figure it out from this point.
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