As an example we consider our old friend the equation

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Unformatted text preview: k=0 6 1 (x − π )2k . 22 k 1.1.3. Use the comparison test and the ratio test to find the radius of convergence of the power series ∞ x 2m 1 (−1)m . 2 (m!) 2 m=0 1.1.4. Determine the values of x0 at which the following functions fail to be real analytic: 1 x a. f (x) = , b. g (x) = 2 , x−4 x −1 4 1 c. h(x) = 4 , d. φ(x) = 3 x − 3x2 + 2 x − 5x2 + 6x 1.2 Solving differential equations by means of power series Our main goal in this chapter is to study how to determine solutions to differential equations by means of power series. As an example, we consider our old friend, the equation of simple harmonic motion d2 y + y = 0, dx2 (1.3) which we have already learned how to solve by other methods. Suppose for the moment that we don’t know the general solution and want to find it by means of power series. We could start by assuming that ∞ y = a0 + a1 x + a2 x2 + a3 x3 + · · · = an xn . (1.4) n=0 It can be shown that the standard technique for differentiating polynomials term by term also works for power series, so we expect that ∞ d...
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