Chap11_Sec9

# 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.
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?
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.
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 .
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.
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.
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
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
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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