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Math Help - Ratio test question

  1. #1
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    Ratio test question

    hi everyone

    need help with these questions.

    Determine the series converges or diverges by using ratio test.

    a) \sum ^\infty _{n=1} \frac {(-10)^n}{n!}
    = \frac {\frac {(-10)^{(n+1)}}{(n+1)!}}{\frac {(-10)^n}{n!}}
    = \frac{-10}{n+1}
    the series converges.

    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}  }

    i'm stuck on how to simplify,

    really appreciate all your help & guidance.

    thank you & regards.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by anderson View Post
    hi everyone

    need help with these questions.

    Determine the series converges or diverges by using ratio test.

    a) \sum ^\infty _{n=1} \frac {(-10)^n}{n!}
    = \frac {\frac {(-10)^{(n+1)}}{(n+1)!}}{\frac {(-10)^n}{n!}}
    = \frac{-10}{n+1}
    the series converges.

    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}  }

    i'm stuck on how to simplify,

    really appreciate all your help & guidance.

    thank you & regards.
    Try the root test for the second one. I'm sure you'll find a pleasant result.
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  3. #3
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    A single little question... does the ratio test works on series of alternating terms?
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  4. #4
    Super Member General's Avatar
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    Quote Originally Posted by anderson View Post
    hi everyone

    need help with these questions.

    Determine the series converges or diverges by using ratio test.

    a) \sum ^\infty _{n=1} \frac {(-10)^n}{n!}
    = \frac {\frac {(-10)^{(n+1)}}{(n+1)!}}{\frac {(-10)^n}{n!}}
    = \frac{-10}{n+1}
    the series converges.

    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}  }

    i'm stuck on how to simplify,

    really appreciate all your help & guidance.

    thank you & regards.
    this "=" shoud not be there.
    Its mean the sum of the series = the value of the limit of the ratio test!
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  5. #5
    Super Member General's Avatar
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    Quote Originally Posted by felper View Post
    A single little question... does the ratio test works on series of alternating terms?
    Yes.
    The absolute value in the limit of the ratio test will cancel the negative signs.
    The same thing for the root test.
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  6. #6
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    Quote Originally Posted by General View Post
    Yes.
    The absolute value in the limit of the ratio test will cancel the negative signs.
    The same thing for the root test.
    Aaahaa, i onle knew the positive terms version.
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