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Math Help - How to simplify this question

  1. #1
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    How to simplify this question

    Determine the series is absolutely convergent or diverges by using ratio test.
    b) \sum ^\infty _{n=1} {(1+\frac{1}{n})}^{n^2}
    \frac {{(1+\frac{1}{n+1})}^{(n+1)^2}}{{(1+\frac{1}{n})}^  {n^2}}
    = \frac {{(1+\frac{1}{n+1})}^{n^2}+{(1+\frac{1}{n+1})}^{2n  }+{(1+\frac{1}{n+1})}^{2}}{{(1+\frac{1}{n})}^{n^2}  }

    {(1+\frac{1}{n+1})}^{2}+{(1+\frac{1}{n+1})}^{2n}
    = e^{2n} +{e^2}

    how do i simplifly this question,please help. realyl appreaciate all your help & support
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  2. #2
    MHF Contributor

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    Quote Originally Posted by anderson View Post
    Determine the series is absolutely convergent or diverges by using ratio test.
    b) \sum ^\infty _{n=1} {(1+\frac{1}{n})}^{n^2}
    \frac {{(1+\frac{1}{n+1})}^{(n+1)^2}}{{(1+\frac{1}{n})}^  {n^2}}
    = \frac {{(1+\frac{1}{n+1})}^{n^2}+{(1+\frac{1}{n+1})}^{2n  }+{(1+\frac{1}{n+1})}^{2}}{{(1+\frac{1}{n})}^{n^2}  }
    This does not follow. I think you are arguing that (x+1)^2= x^2+ 2x+ 1 so a^{(x+1)^2}= a^{x^2+ 2x+ 1}= a^{x^2}+ a^{2x}+ a^1. That last equality is incorrect. That should be a product, not a sum.

    {(1+\frac{1}{n+1})}^{2}+{(1+\frac{1}{n+1})}^{2n}
    = e^{2n} +{e^2}

    how do i simplifly this question,please help. realyl appreaciate all your help & support
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  3. #3
    Member
    Joined
    Feb 2009
    Posts
    130
    Dear HallsofIvy,

    thank you for replying.
    it should be

    Determine the series is absolutely convergent or diverges by using ratio test.
    b) \sum ^\infty _{n=1} {(1+\frac{1}{n})}^{n^2}
    \frac {{(1+\frac{1}{n+1})}^{(n^2+2n+1)}}{{(1+\frac{1}{n}  )}^{n^2}}

    i dont know how should i continue from here ..need some help & guidance for this question...for all i know the final answer is e, am wondering how to get the answer.

    appreciate all your help & support.
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