Numerical approximations

Write the number to be approximated as a convenient constant multiplied by \((1+u)^\alpha\), with \(|u|\) small.

Choose the substitution from the original expression, check validity, then substitute into the truncated expansion. Keep exact fractions until the final numerical evaluation.

The condition \(|u|<1\) justifies the infinite expansion. It does not guarantee that three terms give a specified number of correct decimal places.

When a question asks for an estimate, report it as an approximation. When technology is permitted, compare with the original value to assess its absolute or percentage error.

For positive small \(u\), \((1-u)^{-1}=1+u+u^2+\cdots\) has positive omitted terms, so any finite partial sum underestimates the exact value.