Taylor Series nts.pdf - rc\u0007 rc decreases on some interval[0 and thus rc(\u0006 < r On the other hand under the assumptions listed in(4 the condition 0 \u2264 r

Taylor Series nts.pdf - rcu0007 rc decreases on some...

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r c , r c decreases on some interval [ 0 , ] and thus r c ( ) < r . On the other hand, under the assumptions listed in ( 4 ) , the condition 0 r D is sufficient but not necessary for the conclusion that for some values of p ( 0 , 1 ) , r c > r . For example, for the generalized FPP Problem with p = q = 1 / 2 and thus λ = 1 / 4, the right side of ( 3 ) is 4 /( 81 ) [ 7 + 130 ] . = . 909. Thus if ( 56 / 65 ) < r < 4 /( 81 ) [ 7 + 130 ] , then r c ( 1 / 4 ) > r even though r > D . References 1. D. Freedman, R. Pisani, and R. Purves, Statistics , 3rd ed., W. W. Norton, New York, 1998. On the Remainder in the Taylor Theorem Lior Bary-Soroker ([email protected]), Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel, and Eli Leher ([email protected]), School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel doi:10.4169/074683409X475706 We give a short straightforward proof for a bound on the reminder term in the Taylor theorem. The proof uses only induction and the fact that f 0 implies the mono- tonicity of f , so it might be an attractive proof to give to undergraduate students. Let f be an n -times differentiable function in a neighborhood of a R . Recall that the Taylor polynomial of order n of f at a is the polynomial P n ( x ) = f ( a ) + f ( a )( x a ) + · · · + f ( n ) ( a ) n !