mth.122.handout.20

mth.122.handout.20 - MTH 122 — Calculus II Essex County...

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Unformatted text preview: MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #20 — Sakai Web Project Material 1 Power Series 1. A power series is a series of the form f ( x ) = ∞ X n =0 a n x n = a + a 1 x + a 2 x 2 + a 3 x 3 + ··· where x is a variable and the a n ’s are constants called the coefficients of the series. The domain of this function is the set of all x for which this series is convergent. 2. A power series in ( x- b ) is a power series centered at b , where b is a constant. f ( x ) = ∞ X n =0 a n ( x- b ) n = a + a 1 ( x- b ) + a 2 ( x- b ) 2 + a 3 ( x- b ) 3 + ··· where x is a variable and the a n ’s are constants called the coefficients of the series. The domain of this function is the set of all x for which this series is convergent, and you should notice that this series always converges for x = a . 3. Theorem : For a given power series ∞ X n =0 a n ( x- b ) n = a + a 1 ( x- b ) + a 2 ( x- b ) 2 + a 3 ( x- b ) 3 + ··· there are only three possibilities: (a) The series converges only when x = b . The radius of convergence is defined to be r = 0. (b) The series converges for all x . The radius of convergence is defined to be r = ∞ . (c) There is a positive number R such that the series converges if | x- b | < R and diverges for | x- b | > R . What happens at | x- b | = R should also be examined. The radius of convergence is between b- r and b + r , including any endpoints where the series converges. 4. Theorem: If the power series ∞ X n =0 a n ( x- b ) n = a + a 1 ( x- b ) + a 2 ( x- b ) 2 + a 3 ( x- b ) 3 + ··· 1 This document was prepared by Ron Bannon ( [email protected] ) using L A T E X2 ε . Last revised January 10, 2009. 1 has a radius of convergence R > 0, the the function defined by f ( x ) = a + a 1 ( x- b ) + a 2 ( x- b ) 2 + a 3 ( x- b ) 3 + ··· = ∞ X n =0 a n ( x- b ) n is differentiable (and therefore continuous) on the interval ( b- R, b + R ). 5. Method for Computing the Radius of Convergence: To calculate the radius of convergence, r , for the power series ∞ X n =0 a n ( x- b ) n , use the ratio test....
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This note was uploaded on 01/31/2011 for the course MTH 222 taught by Professor Ban during the Spring '10 term at Essex County College.

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mth.122.handout.20 - MTH 122 — Calculus II Essex County...

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