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Re: Re: Factorial algorithm execution time

by Anonymous Monk
on Oct 26, 2002 at 18:39 UTC ( [id://208260]=note: print w/replies, xml ) Need Help??

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  1. or download this
    log(n!) = log(1) + log(2) + ... + log(n)
        = 1/2 log(1) + (log(1) + log(2))/2
...
                     + ...
                     + (log(n-1) + log(n))/2
                     + 1/2 log(n)
  2. or download this
    log(n!) = 1/2 0 + Integral from 1 to n of log(x) +
          1/2 log(n) + e(1) + e(2) + ... + e(n-1)
  3. or download this
    log(n!) = 1/2 0 + Integral from 1 to n of log(x) +
          1/2 log(n) + e(1) + e(2) + ... + e(n-1)
...
          - (e(n) + e(n+1) + e(n+2) + ...)
        = n*log(n) - n + log(n)/2 + E + 1
          - e(n) - e(n+1) - e(n+2) - ...
  4. or download this
    n! = K * sqrt(n) + n**n * (1 + O(1/n)) / e**n

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