Section10.6-Power Series

4 findpowerseriesrepresentationsforthe

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Unformatted text preview: 1 3 x n 0 3 n for x 1 or x 3 3 TryIt.4 Find power series representations for the following functions, and determine the interval of convergence: 1 1 8 x3 7 2 3x 3 2 5x 2 x3 1 x2 Theorem bn x c If the power series, , has radius of convergence, n n 0 f ( x) bn x c R > 0, then the function, , is differentiable, n n 0 and therefore continuous, on the interval ( c – R, c + R) and i) f '( x) nbn x c n 1 n 1 ii) f ( x)dx A b n0 n x c n 1 n 1 The radii of convergence of the power series in equations i) and ii) are also R, but the interval of convergence might be different from that of the original power series. Interpretation Power series representations of functions can be integrated or differentiated term by term to yield power series for other functions Differentiation Example 1 2 3 n 1 x x x x 1 x 1 d 1 x 1 2 x 3 x 2 nx n 1 dx 1 1 x 2 nx n 1 n 1 R = 1 I = ( ‐1, 1) R = 1 n 1 x n n0 R = 1 Integration Example 1 2 3 n 1 x x x x 1 x 1 x 2 x3 xn 1 xdx A x 2 3 n x n 1 ln 1 x A n 0 n 1 R = 1 I = ( ‐1, 1) R = 1 Now find A… R = 1 Integration Example continued x n 1 ln 1 x A n 0 n 1 R = 1 Substitute x = 0: 0n 1...
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This note was uploaded on 02/24/2014 for the course APMA 1110 taught by Professor Morris during the Fall '11 term at UVA.

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