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Thread: D'Alembert's ratio test

  1. #1
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    D'Alembert's ratio test

    Use D'Alembert's ratio test to determine the convergent of the following series(hint:Find the general term first)

    i) 1 + \frac{2^2}{2!} +\frac {3^3}{3!} + \frac{4^4}{4!}

    the general term: \frac {n^n}{n!}

    so is it:
    \frac {\frac {(n+1)^{(n+1)}}{{n+1}!}}{\frac {n^n}{n!}}

    is it correct? i don't know how to solve the limit...please help me....

    ii) \frac{1}{2} +\frac {2}{3} + \frac{3}{4}+ \frac{4}{5}

    general term
    \frac {n}{n+1}
    is it

    \frac{\frac {n+1}{n+2}}{\frac {n}{n+1}}

    \frac{n^2+2n+2}{n^2+2n}
    limit = 1
    convergent

    iii) \frac{2}{5}+ \frac {2^2}{6}+ \frac {2^3}{7} +\frac {2^4}{8}

    general term : \frac {2^n}{n+4}
    \frac{\frac {2^{n+1}}{n+5}}{\frac {2^n}{n+4}}

    \frac{2n+8}{n+5}

    limit 2

    2>1, divergent

    please help guide & correct me, all help appreciated..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
    Use D'Alembert's ratio test to determine the convergent of the following series(hint:Find the general term first)

    i) 1 + \frac{2^2}{2!} +\frac {3^3}{3!} + \frac{4^4}{4!}

    the general term: \frac {n^n}{n!}

    so is it:
    \frac {\frac {(n+1)^{(n+1)}}{{n+1}!}}{\frac {n^n}{n!}}

    is it correct? i don't know how to solve the limit...please help me....

    ii) \frac{1}{2} +\frac {2}{3} + \frac{3}{4}+ \frac{4}{5}

    general term
    \frac {n}{n+1}
    is it

    \frac{\frac {n+1}{n+2}}{\frac {n}{n+1}}

    \frac{n^2+2n+2}{n^2+2n}
    limit = 1
    convergent

    iii) \frac{2}{5}+ \frac {2^2}{6}+ \frac {2^3}{7} +\frac {2^4}{8}

    general term : \frac {2^n}{n+4}
    \frac{\frac {2^{n+1}}{n+5}}{\frac {2^n}{n+4}}

    \frac{2n+8}{n+5}

    limit 2

    2>1, divergent

    please help guide & correct me, all help appreciated..thank you & regards
    It would be easier to note that \frac{n^n}{n!}\to\infty...but \frac{(n+1)^{n+1}}{(n+1)!}\cdot\frac{n!}{n^n}=\lef  t(1+\frac{1}{n}\right)^n\to?. You should recognize this last limit.
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  3. #3
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    thank you for replyinh.

    is it e?

    is 1 >e, the answer is divergent?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by anderson View Post
    thank you for replyinh.

    is it e?

    is 1 >e, the answer is divergent?
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    Quote Originally Posted by Drexel28 View Post
    thank you so much for helping. just wanted to know if i did the other two questions correctly, can anyone help me to confirm..really appreciate all your help & guidance.

    for ii) the answer is no conclusion because L=1.

    please help feedback.

    thank you & regards
    Last edited by anderson; Feb 11th 2010 at 12:11 AM.
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