Handbook of Differential Equations: Ordinary Differential Equations, Volume 1
Differential Equations Chapter 8 Review ppt
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Transcript of Differential Equations Chapter 8 Review ppt
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DIFFERENTIAL EQUATIONS
Chapter 8 HELP!!
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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.
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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
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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
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8.2 Examples p 434-436
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8.3 Examples p. 445-446
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