DIFFERENTIAL EQUATIONS

Chapter 8 HELP!!

8.1 Introduction to Taylor Polynomial Approximation

- When you need to numerically approximate a function f(x) near a particular point x_0 use the Taylor Polynomial

- This can also be written as

 - We can estimate the accuracy to which the Taylor polynomial approximates

the target function for (where measures the accuracy of the approximation.) 

 - The function above along with the Lagrange gets you the error function below.

 

 

8.2 Power Series and Analytic Functions

- Power Series about the point  

  (Where x is a variable and the are constants.)  

o This Converges at point x = c if (what is listed below converges)  

 o If the limit DNE then the power series is said to Diverge

 

- Theorem 1:

o For each given power series form (1) there will exist a number of

Which converges absolutely if

Which diverges if

o All of this information is called the radius of convergence

o If the series (1) converges for all values of x, then

o When the series (1) converges only at

The RATIO TEST is used to find out what happens at

 

- Theorem from calculus 2 book (Stewart Calculus 6th edition

p.728)

o If the series is convergent, then the

- Divergence test

o If the or if then series diverges

8.2 Examples p 434-436

8.3 Examples p. 445-446