sn10_1 - Expand ∑ ∞ = n n n x a Now what happens when...

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Math 1432 Notes – session 10 11.7 Power Series Let’s look at x x f - = 1 1 ) ( Taylor series for this about 0 is: = 0 n n n x a is a Power Series centered at 0. ( 29 0 n n n a x b = - is a Power Series centered at b . Example: Find the power series for 2 2 4 ) ( x x x f - =
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For the Power Series centered at b , ( 29 0 n n n a x b = - , the radius of convergence is the largest value R so that the power series converges for x b R - < Absolute convergence determines our radius of convergence. Possibilities: ( 29 0 n n n a x b = - converges: 1. 2. 3. 4. 5. 6. Examples: 1. Find the radius and interval of convergence for ( 29 - n n x n 1
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2. Find the radius and interval of convergence for ( 29 ( 29 1 1 1 k k x k - - + 3. Find the radius and interval of convergence for ( 29 ( 29 1 1 2 1 k k k x k - + +
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DAILY EMCF 10 1. When checking the endpoints for the interval of convergence, the series must absolutely converge when the endpoint is plugged in. a. True b. False From the final exam review #14a 1 1 0 ( 2) ( 1)3 n n n x n + + = - +
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11.8 Differentiation and Integration of Power Series
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Unformatted text preview: Expand ∑ ∞ = n n n x a Now, what happens when we take the derivative of this? Thm – If ∑ ∞ = n n n x a converges on (-c, c) then ( 29 ∑ ∞ = n n n x a dx d converges on (-c, c) (you still must check the endpoints for each problem) Example: Show that x x dx d cos sin = using their power functions Integration of Series: Thm – If ∑ ∞ = = ) ( n n n x a x f converges on (-c, c), then ∑ ∞ = + + = 1 1 ) ( n n n x n a x g converges on (-c, c) and C x g dx x f + = ∫ ) ( ) ( More examples: 1. Find a power series for x 1 tan-using integration. 2. Given 2 ( ) cos f x x x = , find ( 29 9 f . 3. Expand ln(cos( )) x in powers of x . DAILY EMCF 10 2. When taking the integral of a series, you don’t need to put +C if its indefinite. a. True b. False Final Exam Review:...
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This note was uploaded on 03/07/2012 for the course MATH 11278 taught by Professor Jeffmorgan during the Summer '10 term at University of Houston.

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sn10_1 - Expand ∑ ∞ = n n n x a Now what happens when...

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