SalasSV_11_06 - 11.6 TAYLOR POLYNOMIALS AND TAYLOR SERIES...

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11.6 TAYLOR POLYNOMIALS AND TAYLOR SERIES IN x a ± 677 ± 11.6 TAYLOR POLYNOMIALS AND TAYLOR SERIES IN x a So far we have encountered series expansions only in powers of x . Here we generalize to expansions in powers of x a , where a is an arbitrary real number. We begin with a more general version of Taylor’s theorem. THEOREM 11.6.1 TAYLOR’S THEOREM If g has n + 1 continuous derivatives on an open interval I that contains the point a , then for each x I , g ( x ) = g ( a ) + g ± ( a )( x a ) + g ±± ( a ) 2 ! ( x a ) 2 +···+ g ( n ) ( a ) n ! ( x a ) n + R n ( x ) where R n ( x ) = 1 n ! ± x a g ( n + 1) ( t )( x t ) n dt . The polynomial P n ( x ) = g ( a ) + g ± ( a )( x a ) + g ±± ( a ) 2 ! ( x a ) 2 +···+ g ( n ) ( a ) n ! ( x a ) n is called the nth Taylor polynomial for g in powers of x a . In this more general setting, the Lagrange formula for the remainder, R n ( x ), takes the form (11.6.2) R n ( x ) = g ( n + 1) ( c ) ( n + 1) ! ( x a ) n + 1 where c is some number between a and x . Now let x I , x ²= a , and let J be the interval that joins a to x . Then (11.6.3) | R n ( x ) |≤ ² max t J | g ( n + 1) ( t ) | ³ | x a | n + 1 ( n + 1) ! . If R n ( x ) 0, then we have the series representation g ( x ) = g ( a ) + g ± ( a )( x a ) + g ±± ( a ) 2 ! ( x a ) 2 +···+ g ( n ) ( a ) n ! ( x a ) n +··· , which, in sigma notation, takes the form (11.6.4) g ( x ) = k = 0 g ( k ) ( a ) k ! ( x a ) k . This is known as the Taylor expansion of g ( x ) in powers of x a . The series on the right is called a Taylor series in x a . All this differs from what you saw before only by a translation. DeFne f ( x ) = g ( x + a ).
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678 ± CHAPTER 11 INFINITE SERIES Then f ( k ) ( x ) = g ( k ) ( x + a ) and f ( k ) (0) = g ( k ) ( a
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SalasSV_11_06 - 11.6 TAYLOR POLYNOMIALS AND TAYLOR SERIES...

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