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chap11sec8 - 11.8 Power Series Manyfamiliar(andunfamiliar)...

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11.8 Power Series Many familiar (and unfamiliar) functions can be written in the form  2 3 0 1 2 3 c c x c x c x + + + + L  as an infinite sum  of the product of certain numbers  n c  and powers of the variable  x . Such expressions are called  power series with center 0; the numbers  n c  are called its coefficients,  0 n n n c x = . Slightly more general, an expression of the form  2 0 1 2 ( ) ( ) c c x a c x a + - + - + L  is called a  power series in (x-a) or a  power series centered at a,   0 ( ) n n n c x a = - . So, the question becomes, "when does the power series converge?" Any of the series tests are available for use, but most often the  Ratio Test is used. In general this will boil down to  1 lim n n n c x a c + →∞ - When this limit is between –1 and 1, the series  converges. There are only 3 possibilities for how this series can converge.

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This note was uploaded on 01/20/2012 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.

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chap11sec8 - 11.8 Power Series Manyfamiliar(andunfamiliar)...

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