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Does the denominator become larger faster

  1. Mar 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Determine if convergent or divergent. Determine limit if convergent.

    2. Relevant equations
    [itex]a_{n} = \frac{n!}{n^n}[/itex]

    3. The attempt at a solution
    As per the hint, i use 1/n to compare.

    however, how is this statement true:

    [itex]\lim_{x\to\infty} \frac{n!}{n^n} <= \lim_{x\to\infty} \frac{1}{n} [/itex]??

    does the denominator become larger faster than the numerator making it a smaller number than the 1/n?
  2. jcsd
  3. Mar 19, 2013 #2


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    Homework Helper

    Think about it. E.g. 3!/3^3 is less than 1/3. Why is that? That's (1*2*3)/(3*3*3).
  4. Mar 21, 2013 #3
    update: nevermind.
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