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Math Help - Converge or Diverge

  1. #1
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    Converge or Diverge

    Here is a nasty series:

    Pn = (n+1)2^n/(n!)^2

    Using the ratio test I get:

    Pn+1/Pn = 2/(n+1)(n!)

    However I dont think this is correct, plus I dont even Know if this is the best way to test if the above nasty is convergent

    and Im going to cry because I'm stuck

    please help, sniff
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by partyshoes View Post
    Here is a nasty series:

    Pn = (n+1)2^n/(n!)^2

    Using the ratio test I get:

    Pn+1/Pn = 2/(n+1)(n!)

    However I dont think this is correct, plus I dont even Know if this is the best way to test if the above nasty is convergent

    and Im going to cry because I'm stuck

    please help, sniff
    using the ratio test is fine. but that's wrong.

    By the ratio test: \lim_{n \to \infty} \left| \frac {P_{n + 1}}{P_n} \right| =  \lim_{n \to \infty} \left| \frac {\frac {(n + 2)2^{n + 1}}{[(n + 1)!]^2}}{\frac {(n + 1)2^n}{(n!)^2}} \right| = \lim_{n \to \infty} \left| \frac {(n + 2)2^{n + 1}}{[(n + 1)!]^2} \cdot \frac {(n!)^2}{(n + 1)2^n} \right|

    now continue
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  3. #3
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    Cancelled out the 2^n but still left with;

    2(n+2)n!^2/(n+1)!^2(n+1)

    What can I do to simplify this?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by partyshoes View Post
    Cancelled out the 2^n but still left with;

    2(n+2)n!^2/(n+1)!^2(n+1)

    What can I do to simplify this?
    what you wrote is confusing me. your syntax is off.

    start by grouping the common things. that is, put the factorial over the factorial, the 2^(whatever) over the 2^(whatever) and so on. you end up with

    \lim_{n \to \infty} \left| \frac {n + 2}{n + 1} \cdot \frac {2^{n + 1}}{2^n} \cdot \left( \frac {n!}{(n + 1)!} \right)^2 \right|

    now can you continue?
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  5. #5
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    Does this simplify to:


    2(n+2)/(n+1)^3
    Last edited by partyshoes; November 26th 2007 at 10:33 AM. Reason: OK Ive got the idea now
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by partyshoes View Post

    Does this simplify to:


    2(n+2)/(n+1)^3
    yes. now what does the limit of that function go to? does it converge?
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