Alternating Series Test

9.9 APPROXIMATIONS BY POWER SERIES

Power series are one of the most important methods of approximation in mathematics. Consider a power series

f(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ + ....

The partial sums give approximate values for the function,

f(x) ~ a₀ + a_xx + a₂x² + ... + a_nxⁿ,

and the tails E_n give the error in the approximation,

f(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ + E_n.

If we can estimate the error E_n we can compute approximate values for f(x) to any desired degree of accuracy.

In this section we shall give two simple methods of estimating the error. A more general method will be given in the next section. Our first method is to use the Alternating Series Test. It can be applied whenever a power series is alternating.

Example 1: ln

Example 2: arctan

Example 3: e