Notes on Second Order Differential Equations. Be able to give the general solution for an equation of the form y'' + c1y' + c0y = 0. Be able to find the particular solution satisfying initial conditions y(0)=y0 , y'(0)=y'0. (Example 3 -both parts-, Example 4 -both parts-, Example 5 -both parts-, Example 6; Exercises 6,8,9.)
8.1 Know examples of convergent and divergent sequences. In case an = f(n) where f is a familiar function, be able to use your knowledge about limx --> infinity f(x) to calculate limn --> infinity an (Theorem 2 p.562; Examples 3,4; Exercises 9-12.)
8.2 Understand the difference between a sequence and a series! Understand that the sum of a series is the limit of the sequence of its partial sums (Definition 2 p.571) and how the arithmetic of finite sums extends to the arithmetic of series (Theorem 8 p.575). Understand the behavior of GEOMETRIC SERIES ( Example 1 <-- VERY IMPORTANT; Examples 2-5; Exercises 3, 13-16.) Understand that the terms can go to zero without the series converging (Example 7 <-- IMPORTANT: HARMONIC SERIES.)
8.3 Understand how to use what you know about convergence or divergence of the integral from 1 to infinity of the function f(x) to show convergence or divergence of the series f(1) + f(2) + f(3) + ... (Integral test, p. 581; Example 1; Example 2 <-- IMPORTANT: p-SERIES; Exercise 1, Exercises 6-10.) Understand that for two positive series, if the larger one converges then the smaller must converge (Comparison test, p.583; Examples 3,4; Exercises 12-16.)
8.4 Understand that an alternating series with terms decreasing to zero must converge, and be able to use the inequality |Limit - nth partial sum| < |(n+1)st term| to estimate the accuracy of the nth partial sum in this case (Examples 2,3,4; Exercises 3,4,5,8; Exercise 12.) Understand how to apply the RATIO TEST and that when the limit of the ratios of consecutive terms is 1, the test is inconclusive (Examples 8,9; Note on p.594; Exercises 19,20,31.)
8.5 Be able to use the Ratio Test to calculate the radius of convergence of a power series. (Examples 1,2; Exercises 5,6,7,8.) The behavior of the series at the endpoints must be determined by additional tests! Be able to do this and to establish the interval of convergence (radius plus or minus left or right-hand endpoint). (Examples 4,5; Exercises above.)
8.6 A power series defines a function f(x) on its interval of convergence. Be able to cook up geometric series for functions like f(x)= 1/(1+x2) and to calculate their intervals of convergence (Examples 1,2; Exercises 3,4,5.) Be able to apply Theorem 2 p.604 to calculate the power series for the derivative and the anti-derivative of f. Remember that these new power series have the same radius of convergence as the original one. (Examples 5,7; Exercises 9,10,19,21.)
8.7 Understand why if f(x)=c0 + c1x + c2x2 + ... then c0 = f(0), c1 = f'(0), ... , cn = f(n)(0)/n! .... Be able to use this fact to calculate power series for ex, sin(x), cos(x). Know how to use Taylor's inequality (Theorem 9 p.611) to calculate how many terms are needed to approximate f(x) to within a given tolerance. (Examples 1,2,4,5; Exercises 4,15,17.) Be able to work with Taylor series centered at a point x=a different from zero (Example 3; Exercises 11,14.)
8.10 Be able to use the power series method to solve differential equations, both those which we can already solve explicitly (like y'=y and y''+y=0) and those which we cannot (non-linear, non-constant coefficients, non-homogeneous). (Examples 1,2; Exercises 1,2,7.)
Use the "Chapter 8 Review" p.637 for further reviewing, especially Exercises 1-46, 51,52.