A rate of 0.25% a month compounds for 48 months. Give 1.0025 to the power 48 to four significant figures in your head, and bound the error of your method.

A rate of 0.25% a month compounds for 48 months. Give 1.0025 to the power 48 to four significant figures in your head, and bound the error of your method.

Approach: Move into logs with a two-term expansion, multiply by the number of periods, then exponentiate with a short power series and size the first omitted term in each.

1.1273. Work in logs with the series expansion ln(1 + x) = x - x^2/2, so ln(1.0025) = 0.0025 - 0.000003125 = 0.00249688, and 48 times that is 0.119850. Exponentiate with the exponential series 1 + y + y^2/2 + y^3/6 at y = 0.11985: 1 + 0.11985 + 0.007182 + 0.000287 = 1.127319, and the fourth term adds 0.0000086 to give 1.12733. The linear answer of 48 * 0.25% = 12% understates the total by 0.73 percentage points, which is exactly the compounding a linear estimate drops. The error bound comes from the first omitted term in each series, 0.0025 cubed over 3 in the log and y^5/120 in the exponential, both far below 10^{-6}, so four significant figures are safe.

Follow-up: What monthly rate compounds to exactly 1.15 over 48 months, and how would you invert the same two series to find it?

Key concepts: series expansion, compounding, error bound, exponential.