BC ONLYExample 50 __
Find the third-degree Maclaurin polynomial for f(x) = cos , 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.
Thus .
Since this is not an alternating series for x = 0.1, we must use the Lagrange error
bound:
Note that is decreasing on the interval 0 < c < 0.1, so its
maximum value occurs at c = 0. Hence:
C6. Computations with Power Series
The power series expansions of functions may be treated as any other functions
for values of x that lie within their intervals of convergence. They may be added,
subtracted, multiplied, divided (with division by zero to be avoided),
differentiated, or integrated. These properties provide a valuable approach for
many otherwise difficult computations. Indeed, power series are often very
useful for approximating values of functions, evaluating indeterminate forms of
limits, and estimating definite integrals.