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Simple limit question

  1. Apr 12, 2005 #1
    How do I show that [tex] \mathop {\lim }\limits_{n \to \infty } \left( {n!!} \right)^{\left( {n^{ - n} } \right)} = 1 [/tex] ?

    (The n^-n forces the value to decrease faster than n!! increases, I believe. But how to work out that?)
     
    Last edited: Apr 12, 2005
  2. jcsd
  3. Apr 12, 2005 #2
    1)Firstly we'll let the expression (n!!)^(n^-n)=y, then taking ln on both sides give,
    lny=ln(n!!)/n^n.
    2) We'll then find the limit of the expression at the RHS using Le Hopital's rule or by observation that n^n increases faster than ln(n!!). =)
    3) After finding the limit, L, all we have to do is substitute back the value of y, which is e^L
     
    Last edited: Apr 12, 2005
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