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sec11_8 - is one of the intervals a-R,a R a-R,a R a-R,a R...

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11.8 Power Series 1. Power Series n =0 c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + · · · Example 1.1. n =0 x n = 1 + x + x 2 + x 3 + · · · Definition 1.1. A series of the form n =0 c n ( x - a ) n where c n , and a are constants and x is a variable, is called a power series in ( x - a ) , or centered at a , or about a . Remark 1.1. A typical question: For what values of x is the series convergent? Example 1.2. For what values of x is the series convergent? n =0 n ! x n 1
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Section 11.8 Power Series 2 Theorem 1.1. For a given power series c n ( x - a ) n , there are three possibilities. (1) The series converges for x = a only. In this case we say the radius of convergence of the series is R = 0 and the interval of convergence is { a } . (2) The series converges for all x . In this case we say the radius of convergence of the series is R = and the interval of convergence is ( -∞ , ) . (3) There is a positive real number R where the series converges for all x with | x - a | < R and diverges for all x with | x - a | > R . In this case we say the radius of convergence of the series is R . The
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Unformatted text preview: is one of the intervals ( a-R,a + R ) , ( a-R,a + R ] , [ a-R,a + R ) , or [ a-R,a + R ] . Remark 1.2. Knowing the radius of convergence is a positive real number, R , does not tell you whether the series converges or diverges when | x-a | = R . Example 1.3 (Example of a Bessel Function. These equations ±rst arose in solv-ing Kepler’s equation to describe planetary motion) . Find the radius and interval of convergence. J ( x ) = ∞ ± n =0 (-1) n x 2 n 2 2 n ( n !) 2 Section 11.8 Power Series 3 Example 1.4. (problem 6) Find the radius and interval of convergence. ∞ ± n =1 √ nx n Example 1.5. (problem 18) Find the radius and interval of convergence. ∞ ± n =1 n 4 n ( x + 1) n Example 1.6. (problem 20) Find the radius and interval of convergence. ∞ ± n =1 (3 x-2) n n · 3 n...
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