 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/7/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Separate the even factors from

1000!

1000! = (1 \cdot 3 \dots 999) (2 \cdot 4 \dots 1000) = 999!! 2^{500} 500! .

The double factorial therefore satisfies

ln (999!!) = ln (1000!) - 500 ln 2 - ln (500!)

. Applying the same fit consistently gives

ln (1 \cdot 3 \dots 999) \approx L (1000) - 500 ln 2 - L (500) ≃ 2954.2213.

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