N! = NN (1-xi) (3) П for values of xi from 0 to 1/N. Taking the natural logs of this we have Log (N!) = N log (N) + log (1 xi) (4) ∑ – for the range xi from 0 to 1/N as i goes from 0 to N-1 The sum in equation 4 cannot be evaluated directly but we can approximate it by replacing xi with a continuous variable x so that log (1-xi) = log (1-x) d x (5) ∑ ∫ Noting that N d i = d x then leads to our approximate expression: Log (N!) = N log (N) + N log (1-x) d x (6) ∫ where the limits are 1/N and 1. The integral can be evaluated by an integration by parts following a change of the variable to leading to: Log (N!) = N log (N) - N + log (N) (7) For large N, log (N) is negligible compared to N, and we have equation given above. Stirling’s It is natural to ask how our expression compares with series expansion (1): Stirling’s Log (N!) = N log (N) N (8) – +0.5log (2πN) +1/ (12N) Comparison of the values of the extra terms in equations 7 and 8 shows they are both negligible (≤ 0.01%) for N The ratio of the values of the extra terms in equation 7 to those in equation 8 is ≥ 100000.
N! = NN П (1-xi) (3) for values of xi from 0 to 1/N. Taking the natural logs of this we have Log (N!) = N log (N) + ∑ log (1 – xi) (4) for the range xi from 0 to 1/N as i goes from 0 to N-1 The sum in equation 4 cannot be evaluated directly but we can approximate it by replacing xi with a continuous variable x so that ∑ log (1-xi) = ∫ log (1-x) d x (5) Noting that N d i = d x then leads to our approximate expression: Log (N!) = N log (N) + N ∫ log (1-x) d x (6) where the limits are 1/N and 1. The integral can be evaluated by an integration by parts following a change of the variable to leading to: Log (N!) = N log (N) - N + log (N) (7) For large N, log (N) is negligible compared to N, and we have Stirling’s equation given above. It is natural to ask how our expression compares with Stirling’s series expansion (1): Log (N!) = N log (N) – N +0.5log (2πN) +1/ (12N) (8) Comparison of the values of the extra terms in equations 7 and 8 shows they are both negligible (≤ 0.01%) for N ≥ 100000. The ratio of the values of the extra terms in equation 7 to those in equation 8 is
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