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IntrodPowerSeries

# IntrodPowerSeries - Introduction and Review of Power Series...

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Unformatted text preview: Introduction and Review of Power Series Definition : A power series in powers of x – a is an infinite series of the form c n (x ! a) n = c + c 1 (x ! a) + c 2 (x ! a) 2 + ... + c n (x ! a) n + .... n = " # If a = 0, this is a power series in x: c n x n = c + c 1 x + c 2 x 2 + ... + c n x n + .... n = ! " Definition : A power series c n x ! a ( ) n n = " # is said to converge at a point x if lim p !" c n (x # a) n n = p \$ exists for that x. This series certainly converges for x = a. If may converge for all values of x, or it may converge for some values of x and not for others. Definition : A power series c n x ! a ( ) n n = " # is said to converge absolutely at a point x if the series c n x ! a ( ) n n = " # = c n x ! a n n = " # converges. Remark : A series that converges absolutely also converges, but the converse it is not necessarily true. The Ratio Test One of the most useful test for absolute convergence of a power series is the ratio test. If c n ≠ 0, and if for a fixed value of x, lim n !" c n + 1 x # a ( ) n + 1 c n x # a ( ) n = x # a lim n !" c n + 1 c n = x # a L then the power series converges absolutely at that value of x if |x – a| L < 1 and diverges if |x – a| L > 1. If |x – a| L = 1, the test is inconclusive. Example : Find the values of x where the power series converges ( ! 1) n + 1 n x ! 2 ( ) n n = " # Applying the ration test, we have lim n !" ( # 1) n + 1 (n + 1)(x # 2) n + 1 ( # 1) n n(x # 2) n = lim n !" x # 2 n + 1 n = x # 2 lim n !" 1 + 1 n \$ % & ’ ( ) = x # 2 According to the ratio test the series converges absolutely if |x – 2| < 1, or 1 < x < 3, and diverges if |x – 2| > 1....
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IntrodPowerSeries - Introduction and Review of Power Series...

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