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ODE Lecture Notes Section 5.1 Page 1 of 6 Section 5.1: Review of Power Series Big Idea: In chapter 5, we will derive power series solutions to second-order differential equations. Thus, some review is in order. Big Skill: You should remember your calc 2 skills related to power series. Review: 1. Definition of convergence of a power series: A power series a x x n 0 m n n 0 is said to converge at a point x if lim an x x0 exists for that x. m n n 0 2. Definition of absolute convergence of a power series: A power series a x x n 0 said to converge absolutely at a point x if a x x n 0 n 0 n n 0 n is converges. a. Absolute convergence implies convergence… 3. Ratio test: If, for a fixed value of x, lim n an 1 x x0 an x x0 n 1 n x x0 lim n an 1 x x0 L , then an the power series converges absolutely at that value of x if x x0 L 1, and diverges if x x0 L 1. If x x0 L 1, then the test is inconclusive. Practice: n Find the values of x for which n x n converges. n0 2 ODE Lecture Notes 4. If a x x n 0 n Section 5.1 Page 2 of 6 converges at x = x1, it converges absolutely for x x0 x1 x0 , and if n 0 it diverges at x = x1, it diverges for x x0 x1 x0 . 5. Radius and interval of convergence: The radius of convergence is a nonnegative number such that a x x n 0 n 0 n converges absolutely for x x0 and diverges for x x0 . a. Series that converge only when x = x0 are said to have = 0. b. Series that converge for all x are said to have = . c. If > 0, then the interval of convergence of the series is x x0 . Practice: Find the radius of convergence of Given that x 1 n 0 n 2n n . an x x0 and bn x x0 converge for x x0 … n n 0 n n 0 6. Sum of series: a x x b x x a n 0 n n 0 n 0 n n 0 n 0 n bn x x0 n ODE Lecture Notes Section 5.1 Page 3 of 6 7. Product and Quotient of series: n n a x x 0 n bn x x0 n 0 n 0 2 3 2 3 a0 a1 x x0 a2 x x0 a3 x x0 ... b0 b1 x x0 b2 x x0 b3 x x0 ... a0 b0 b1 x x0 b2 x x0 b3 x x0 ... 2 3 a1 x x0 b0 b1 x x0 b2 x x0 b3 x x0 ... 2 a x x b b x x b x x 3 b x x ... a2 x x0 b0 b1 x x0 b2 x x0 b3 x x0 ... 2 3 3 0 0 1 0 2 0 2 2 3 3 3 0 ... a0b0 a0b1 a1b0 x x0 a0b2 a1b1 a2b0 x x0 a0b3 a1b2 a2b1 a3b0 x x0 ... 2 3 n n ak bn k x x0 n 0 k 0 To do a quotient, can write as a multiplication and equate terms: a x x n 0 n b x x n 0 n n 0 n n n n n d n x x0 d n x x0 bn x x0 an x x0 n 0 n 0 n 0 n 0 0 OR can do long division… Practice: Find the first few terms of the tangent function using the series for the sine and cosine functions. ODE Lecture Notes Section 5.1 Page 4 of 6 ODE Lecture Notes Section 5.1 Page 5 of 6 8. Derivatives of a series: d n n 1 a x x nan x x0 n 0 dx n 0 n 1 2 d n n2 a x x n n 1 an x x0 0 2 n dx n 0 n2 9. Taylor series: f n x0 n x x0 is the Taylor series for a function f(x) about the point x = x0. n! n 0 10. Equality of series: every corresponding term is equal… 11. Analytic functions: have a convergent Taylor series with non-zero radius of convergence about some point x = x0. ODE Lecture Notes 12. Shift of index of Summation: Section 5.1 Page 6 of 6