Chap11_Sec9 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES
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11.9 Representations of Functions as Power Series INFINITE SEQUENCES AND SERIES In this section, we will learn: How to represent certain functions as sums of power series.
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FUNCTIONS AS POWER SERIES We can represent certain types of functions as sums of power series by either: Manipulating geometric series Differentiating or integrating such a series
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FUNCTIONS AS POWER SERIES You might wonder: Why would we ever want to express a known function as a sum of infinitely many terms?
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FUNCTIONS AS POWER SERIES We will see that this strategy is useful for: Integrating functions without elementary antiderivatives Solving differential equations Approximating functions by polynomials
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FUNCTIONS AS POWER SERIES Scientists do this to simplify the expressions they deal with. Computer scientists do this to represent functions on calculators and computers.
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FUNCTIONS AS POWER SERIES We start with an equation we have seen before: Equation 1 2 3 0 1 1 ... 1 1 n n x x x x x x = = + + + + = < -
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FUNCTIONS AS POWER SERIES We first saw this equation in Example 5 in Section 11.2 We obtained it by observing that it is a geometric series with a = 1 and r = x .
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FUNCTIONS AS POWER SERIES However, here our point of view is different. We regard Equation 1 as expressing the function f ( x ) = 1/(1 – x ) as a sum of a power series.
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FUNCTIONS AS POWER SERIES A geometric illustration of Equation 1 is shown.
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FUNCTIONS AS POWER SERIES Since the sum of a series is the limit of the sequence of partial sums, we have where s n ( x ) = 1 + x + x 2 + … + x n is the n th partial sum. 1 lim ( ) 1 n n s x x →∞ = -
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FUNCTIONS AS POWER SERIES Notice that, as n increases, s n ( x ) becomes a better approximation to f ( x ) for –1 < x < 1.
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FUNCTIONS AS POWER SERIES Express 1/(1 + x 2 ) as the sum of a power series and find the interval of convergence. Example 1
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FUNCTIONS AS POWER SERIES Replacing x by – x 2 in Equation 1, we have: 2 2 2 0 2 2 4 6 8 0 1 1 1 1 ( ) ( ) ( 1) 1 ... n n n n n x x x x x x x x = = = + - - = - = - = - + - + - Example 1
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FUNCTIONS AS POWER SERIES Since this is a geometric series, it converges when |– x 2 | < 1, that is, x 2 < 1, or | x | < 1. Hence, the interval of convergence is (–1, 1). Example 1
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FUNCTIONS AS POWER SERIES Of course, we could have determined the radius of convergence by applying the Ratio Test. However, that much work is unnecessary here. Example 1
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Find a power series representation for 1/( x + 2). We need to put this function in the form
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap11_Sec9 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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