EXAMPLE 49
Find the Maclaurin series for ln (1 + x) and the associated Lagrange error bound.
SOLUTION:
Then
where the Lagrange error bound is
NOTE: For 0 < x < 1 the Maclaurin series is alternating, and the error bound simplifies to
the first omitted term. The more difficult Lagrange error bound applies for −1 < x <0.
EXAMPLE 50
Find the third-degree Maclaurin polynomial for and determine the upper bound on
the error in estimating f (0.1).
SOLUTION: We first make a table of the derivatives, evaluated at x = 0 and giving us the
coefficients.