power series & radius of convergence
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Transcript of power series & radius of convergence
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Example 1
Example 1 – Solution
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Taylor and Maclaurin Series
Example 2
Example 2 – Solution
Taylor and Maclaurin Series
Example 8
Example 8 – Solution
Example 8 – Solution
DEFINATION
A power series about x=0 is a series of the form
=+++…….++……… A power series about x=a is a series of the form
=+++….++…. In which the center a and the coefficieants , , ,…., are constants.
THE CONVERGENCE THEOREM FOR POWER SERIES
If the power series =+++……. Converges for x=c ≠0,then it converges absolutely for all x with |x|<|c|. If the series diverges for x=d , then it diverges for all x with |x|>|d|..
RADIUS OF CONVERGENCE In previous explainations there is a number R so that power
series will converge for , |x – a|< R and will diverge for |x – a|> R.This number is called the radius of convergence for the series.Note that the series may or may not converge if |x – a| =R .What happens at these points will not change the radius of cnvergencce.