Review earlier material using Midterm 1 Review and Midterm 2 Review.

Notes on Second Order Differential Equations. Be able to give the
general solution for an equation of the form `y'' + c _{1}y'
+ c_{0}y = 0`. Be able to find the particular solution satisfying
initial conditions

8.1 Know examples of convergent and divergent sequences. In case
`a _{n} = f(n)` where

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 - n ^{th} partial sum| < |(n+1)^{st} term|` to estimate
the accuracy of the n

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+x ^{2})`
and to calculate their intervals of convergence (

8.7 Understand why if `f(x)=c _{0} + c_{1}x
+ c_{2}x^{2} + ...` then

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.

Math Dept SUNY Stony Brook

tony@math.sunysb.edu

December 11 2000