Stirling’s Approximation Calculator – n! Estimate

Stirling’s formula estimates factorials without multiplying anything: n! ≈ √(2πn)(n/e)ⁿ. For n = 10: √(62.83) ≈ 7.927, (10/e)¹⁰ ≈ 4.5400×10⁵, product ≈ 3,598,696 — within 0.83% of the exact 3,628,800, matching the leading error term 1/(12n) = 1/120 = 0.83%. The approximation is asymptotic: relative error shrinks as n grows (0.083% at n = 100, where the true value has 158 digits) yet the absolute gap widens — estimates are always slightly low without corrections. The calculator prints the two factors, the estimate, the exact value when computable, the relative error, and the first correction (1 + 1/(12n)) which pushes n = 10 to within 0.007%. Uses: log-gammas in statistics, entropy and combinatorics back-of-envelope work (52! ≈ 8.06×10⁶⁷), and anywhere huge factorials meet logarithms via ln n! ≈ n ln n − n.

1. 1

   Enter n

   Any positive integer — large n welcome

2. 2

   Evaluate the two factors

   √(2πn) and (n/e)ⁿ are shown separately

3. 3

   Multiply for the estimate

   3,598,696 for n = 10

4. 4

   Measure the error

   Relative gap and the 1/(12n) prediction compared

• 1

  Relative error ≈ 1/(12n): 0.83% at 10, 0.083% at 100

• 2

  Estimates sit below the true value until corrected

• 3

  Log form: ln n! ≈ n ln n − n + ½ln(2πn)

• Forgetting the √(2πn) prefactor and using only (n/e)ⁿ

• Expecting absolute error to shrink — only relative error does

• Applying it to small n and ignoring the 8%+ error at n = 1

• Mixing e ≈ 2.71828 with 10 in the base