Chapter 4(C) - Power Series Power Series about x = 0 A...

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Power Series
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Power Series about x = 0 01 where , , , , are constants while is a variable. n c c cx LL A power series can be regarded as a function of where it converges. x A about 0 is a series of the form x = power series 2 0 12 0 nn n c x c c x c x = =++ + ++
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21 The geometric series converges to the sum if | | 1 1 and it diverges if 1. n a a r a r ar a r r r - ++ + ++ < - LL Power Series about x = 0 - Example 2 0 1 nn n x x xx = =+++++ 1 This power series about 0 converges to 1 when 1. x x x = - < 2 1 1 , 1 1. 1 n x x x =++++ + -<< - a rx == 1 11 a = -- We state this as Binomial expansion
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Power Series about x = a The number is called the centre of the power series. a More generally, a about is a series of the form xa = power series 2 0 12 0 ( ) ( )() ( ) n n n n n cx a c cxa cxa cxa = - = + - + - ++ -+ L L
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The number is called the centre of the power series. a More generally, a about is a series of the form xa = power series 2 0 12 0 ( ) ( )() ( ) n n n n n cx a c cxa cxa cxa = - = + - + - ++ -+ L L 01 where , , , , are constants while is a variable. n c c LL A about 0 is a series of the form x = power series 2 0 0 nn n c x c c x c x = =++ + ++ Take 0 a =
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2 0 1 nn n x x xx = =+++++ LL 2 1 1 , 1 1. 1 n x x x =++++ + -<< - Put 2 x = 1 Left hand side 1 1 12 1 x = - = - =- 2 Right hand side 1 1248 ... 0 n x Left hand side and Right hand side are not consistent!!
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2 0 1 nn n x x xx = =+++++ LL 2 1 1 , 1 1. 1 n x x x =++++ + -<< - Put 3 x =- 1 Left hand side 1 1 1 ( 3) 1 4 x = - = -- = 2 Right hand side 1 139 2 7 ...
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Chapter 4(C) - Power Series Power Series about x = 0 A...

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