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Bekki George Lecture notes 14

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Unformatted text preview: x − b) n =0 n n ''converges:' ' 1.' ' 2.' ' 3.' ' 4.' ' 5.' ' 6.''' ' # ' Examples:' 1.''Find'the'radius'and'interval'of'convergence'for' ∑ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' (− 1)n n xn ' 2.''Find'the'radius'and'interval'of'convergence'for' ∑ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 3.'Find'the'radius'and'interval'of'convergence'for' ∑ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ( −1) k k +1 ( −1) ( x − 1) k k ⋅ 2 +1 k k ' ( x + 1) k ' ' LecPop14_2 ' ∞ 3. Give'the'radius'of'convergence'for' ∑ x n n !' n=0 ' a. 1 b. 2 c. ½ ' 4. Find the interval of convergence. a) ' ' + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + b) c) e. ∞' d. 0 d) e) ' 11.8++Differentiation+and+Integration+of+Power+Series' ' ∞ Expand' ∑ a n x n ' n =0 ' ' Now,'what'happens'when'we'take'the'derivative'of'this?' ' ' ' ' ' ' ∞ ∞ d Thm'–'If' ∑ a n x n converges on (-c, c) then ∑ a n x n converges on (-c, c) n =0 n = 0 dx (you still must check the endpoints for each problem) ( Example: Show that d sin x = cos x using their power functions dx ) Integration of Series: ∞ ∞ a n n +1 x converges on n =0 n + 1 Thm – If f ( x) = ∑ a n x n converges on (-c, c), then g ( x) = ∑ n =0 (-c, c) and ∫ f ( x)dx = g ( x) + C ' ' More'examples:' 1.''Find'a'power'series'for tan −1 x using'integration.' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2.''Given' f ( x) = x cos x 2 ,'find' f 9 ( 0 ) .' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 3.''Expand ln(cos( x)) in powers of x. ' LecPop14_2 d ∞ xk 5. Find ∑ dx k =0 k + 1 a. b. c. kx k −1 ∑ ( k + 1) k =0 ∞ ∞ kx k ∑ ( k + 1) k =0 ∞ xk ∑ k ( k + 1) k =0 x k +1 ∑ ( k + 1)2 k =0 e. none of these ∞ d....
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