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Thread: 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) $\displaystyle \sum ^\infty _{n=1}$$\displaystyle {(1+\frac{1}{n})}^{n^2}$
    $\displaystyle \frac {{(1+\frac{1}{n+1})}^{(n+1)^2}}{{(1+\frac{1}{n})}^ {n^2}}$
    =$\displaystyle \frac {{(1+\frac{1}{n+1})}^{n^2}+{(1+\frac{1}{n+1})}^{2n }+{(1+\frac{1}{n+1})}^{2}}{{(1+\frac{1}{n})}^{n^2} }$

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

    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) $\displaystyle \sum ^\infty _{n=1}$$\displaystyle {(1+\frac{1}{n})}^{n^2}$
    $\displaystyle \frac {{(1+\frac{1}{n+1})}^{(n+1)^2}}{{(1+\frac{1}{n})}^ {n^2}}$
    =$\displaystyle \frac {{(1+\frac{1}{n+1})}^{n^2}+{(1+\frac{1}{n+1})}^{2n }+{(1+\frac{1}{n+1})}^{2}}{{(1+\frac{1}{n})}^{n^2} }$
    This does not follow. I think you are arguing that $\displaystyle (x+1)^2= x^2+ 2x+ 1$ so $\displaystyle a^{(x+1)^2}= a^{x^2+ 2x+ 1}= a^{x^2}+ a^{2x}+ a^1$. That last equality is incorrect. That should be a product, not a sum.

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

    how do i simplifly this question,please help. realyl appreaciate all your help & support
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
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    Dear HallsofIvy,

    thank you for replying.
    it should be

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

    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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