Definitions: denom(b) = (b^b - b^2 + b - 1) / (b - 1)^2 num(b) = (b^b(b - 2) + 1) / (b - 1)^2

Exact relation: num(b) - (b - 2)denom(b) = b - 1

Therefore: num(b) / denom(b) = (b - 2) + (b - 1)^3 / (b^b - b^2 + b - 1) [exact]

Geometric expansion: Let a = b^2 - b + 1. 1 / (b^b - b^2 + b - 1) = (1 / b^b) * 1 / (1 - a / b^b) = (1 / b^b) * sum_{k>=0} (a / b^b)^k

So: num(b) / denom(b) = (b - 2) • (b - 1)^3 / b^b • (b - 1)^3 * a / b^{2b} • (b - 1)^3 * a^2 / b^{3b} • …

Practical approximation: num(b) / denom(b) ≈ (b - 2) + (b - 1)^3 / b^b

Exact error: Let T_exact = (b - 1)^3 / (b^b - b^2 + b - 1) Let T_approx = (b - 1)^3 / b^b

Absolute error: T_exact - T_approx = (b - 1)^3 * (b^2 - b + 1) / [ b^b * (b^b - b^2 + b - 1) ]

Relative error: (T_exact - T_approx) / T_exact = (b^2 - b + 1) / b^b

Sign: The approximation with denominator b^b underestimates the exact value.

Digit picture in base b: (b - 1)^3 has base-b digits (b - 3), 2, (b - 1). Dividing by b^b places those three digits starting b places after the radix point.

Examples: base 10: 8 + 9^3 / 10^10 = 8.0000000729 base 9: 7 + 8^3 / 9^9 = 7.000000628 in base 9 base 8: 6 + 7^3 / 8^8 = 6.00000527 in base 8

num(b) / denom(b) equals (b - 2) + (b - 1)^3 / (b^b - b^2 + b - 1) exactly. Replacing the denominator by b^b gives a simple approximation with relative error exactly (b^2 - b + 1) / b^b.