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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) = a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n} + .... The partial sums give approximate values for the function, f(x) ~ a_{0} + a_{x}x + a_{2}x^{2} + ... + a_{n}x^{n}, and the tails E_{n} give the error in the approximation, f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n} + 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.


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