Barrons AP Calculus - David Bock

Thus

Since this is not an alternating series for x = 0.1, we must use the Lagrange error bound:

where x = 0.1 and 0 < c < 0.1.

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.

EXAMPLE 51

Compute to four decimal places.

SOLUTION: We can use the Maclaurin series,

and let to get

Note that, since this series converges by the Alternating Series Test, R 5 is less than the first term

dropped: