Chapter 5 Slides Complete ppt

# Chapter 5 Slides Complete ppt - CHAPTER 5 SERIES SOLUTION...

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01/17/11 MAE-182A Lecture Notes Chatterje 1 CHAPTER 5: SERIES SOLUTION OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS POWER SERIES: DEFINITION AND PROPERTIES Power Series Representation of a given function ( ) f x If, for a given x , the limit ( 29 0 0 lim m n n m n a x x →∞ = - exists (bounded) and is equal to ( ) f x , then the series is said to be power series expansion of ( ) f x . Obviously 0 0 ( ) a f x =

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01/17/11 MAE-182A Lecture Notes Chatterje 2 Absolute Convergence of a Power Series A power series is said to converge absolutely at a point x if the series ( 29 ( 29 0 0 0 0 m m n n n n n n a x x a x x = = - = - ∑ ∑ converges If a series converges absolutely at x , then it also converges at x Radius of Convergence of a Power Series (PS) If a power series about 0 x x = converges for all values of 0  in  x x x ρ - < , then is said to be radius of convergence of the PS
01/17/11 MAE-182A Lecture Notes Chatterje 3 Determine ρ for a Given Power Series can be determined from the ratio test ( 29 ( 29 1 1 1 1 0 1 0 0 1 1 0 lim lim lim lim 1 . Hence  n n n n n n n n n n n n n n a a a a a x x a x x a a x x x x + + →∞ + + + →∞ - = - < - - < =

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01/17/11 MAE-182A Lecture Notes Chatterje 4 PROPERTIES OF POWER SERIES CONVERGENCE
01/17/11 MAE-182A Lecture Notes Chatterje 5 EXAMPLE

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01/17/11 MAE-182A Lecture Notes Chatterje 6 7(265)8(5.3)
01/17/11 MAE-182A Lecture Notes Chatterje 7 7(265)8(5.3 )

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01/17/11 MAE-182A Lecture Notes Chatterje 8 FIND RADIUS OF CONVERGENCE WHEN THE SERIES IS GIVEN 7(249)8(5.1)
01/17/11 MAE-182A Lecture Notes Chatterje 9 EQUALITY OF TWO POWER SERIES

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01/17/11 MAE-182A Lecture Notes Chatterje 10 TERM BY TERM DIFFERENTIATION OF A POWER SERIES TAYLOR’S EXPANSION When a poser series is absolutely convergent within a domain bounded by the radius of convergence, the power
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Chapter 5 Slides Complete ppt - CHAPTER 5 SERIES SOLUTION...

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