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# 11-7lect - SECTION 11.7 POWER SERIES A series of the type...

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SECTION 11.7: POWER SERIES A series of the type (1) n =0 a n x n is called a power series. Here a n are fixed real numbers, while x is a variable. a n is the coefficient of x n . The nature of the series (1) is clearly influenced by the particular value of x . The set of those x for which the power series converges is called the domain of convergence . We denote it by S . It is also called the interval of convergence (since it turns out to be an interval). Example: the power series n =0 x n = 1 + x + x 2 + x 3 + . . . is the well known geometric series which converges only for - 1 < x < 1. In other words, in this particular case, S = ( - 1 , 1). We also know the sum of this series when x S , namely 1 x - 1 . Example: n =0 ( - 1 2 ) n = 1 1+1 / 2 = 3 2 ( x = - 1 2 ) n =0 2 n is divergent ( x = 2). Theorem 1 (Fundamental Theorem- J. Hadamard) . Given a power series n =0 a n x n , there exists a number R (radius of convergence) such that the power series is: Absolutely convergent for - R < x < R Divergent for | x | > R The endpoints x = - R and x = R need further investigation.

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11-7lect - SECTION 11.7 POWER SERIES A series of the type...

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