On the remainder in the Taylor theorem
aa r X i v : . [ m a t h . C A ] D ec ON THE REMAINDER IN THE TAYLOR THEOREM
LIOR BARY-SOROKER AND ELI LEHER
Abstract.
We give a short straightforward proof for the bound of the reminder term in theTaylor theorem. The proof uses only induction and the fact that f ′ ≥ f , so it might be an attractive proof to give to undergraduate students. Introduction
Let f be an n -times differentiable function in a neighborhood of a ∈ R . Recall that the Taylorpolynomial of order n of f at a is the polynomial P n ( x ) = f ( a ) + f ′ ( a )( x − a ) + · · · + f ( n ) ( a ) n ! ( x − a ) n . It will be convenient to define P − ( x ) = 0. Let R n = f − P n be the remainder term. Then Theorem 1 (Lagrange’s formula for the remainder) . If f has an ( n + 1) th derivative in [ a, b ] thenthere is some a ≤ ξ ≤ b such that R n ( b ) = f ( n +1) ( ξ )( n + 1)! ( b − a ) n +1 . This formula is the main tool for bounding the remainder term of the Taylor expansion incalculus classes, especially when this subject is taught before integration. Therefore, one wouldlike to have some “natural” proof for it. In [3] it is suggested that induction seems suitable, since P ′ n is the Taylor polynomial of f ′ of order n −
1, hence R ′ n ( x ) is given by induction. The reasonthat this approach fails is that one cannot integrate R ′ n ( x ), since ξ = ξ ( x ) is implicit.While we were teaching a first calculus course for chemistry and physics majors, we observedthat this obstacle can be removed if we slightly change the problem to finding a bound of theremainder, which is all that is needed in order to show that the Taylor series does in fact convergeto the function. From our personal experience, it seems that this approach enables students tograsp the material more easily. Finally we mark that Lagrange’s formula can be deduced from thebound, as we show at the end of this note.The only fact needed in the proof is that a function with a positive derivative is increasing.This can be easily proved with the mean value theorem or without it (see [1, 2]). As a directcorollary one gets: Lemma 2.
Let f, g be differentiable in a closed segment [ a, b ] . If f ( a ) = g ( a ) and f ′ ( x ) ≤ g ′ ( x ) for every x ∈ ( a, b ) , then f ( x ) ≤ g ( x ) for every a ≤ x ≤ b .Proof. Take h = g − f . Thus h ′ ≥ h ( a ) = 0, hence 0 ≤ h ( x ), i.e. f ( x ) ≤ g ( x ). (cid:3) The main result
Theorem 3.
Suppose that f has an ( n + 1) th derivative in [ a, b ] and that m ≤ f ( n +1) ( x ) ≤ M forevery a < x < b . Then for any a ≤ x ≤ b (1) m ( n + 1)! ( x − a ) n +1 ≤ R n ( x ) ≤ M ( n + 1)! ( x − a ) n +1 . Date : Dec. 24, 2008.2000
Mathematics Subject Classification.
Key words and phrases.
Taylor polynomial, Remainder term.
Proof.
By induction on n . For n = − n ≥ f ( x ) = P n ( x ) + R n ( x ). Then f ′ ( x ) = P ′ n ( x ) + R ′ n ( x ). Note that P ′ n is theTaylor polynomial of f ′ of order n − f ′ ) ( n ) = f ( n +1) . Hence by induction we have(2) mn ! ( x − a ) n ≤ R ′ n ( x ) ≤ Mn ! ( x − a ) n , for every a ≤ x ≤ b . Hence Lemma 2 gives the required inequality (since (cid:16) ( x − a ) n +1 ( n +1)! (cid:17) ′ = ( x − a ) n n ! ). (cid:3) We conclude with a proof of Lagrange’s classical formula. As mentioned before, this might beomitted in calculus classes.
Proof.
Choose m = inf a ≤ x ≤ b { f ( n +1) ( x ) } and M = sup a ≤ x ≤ b { f ( n +1) ( x ) } (if f ( n +1) is unbounded, weallow m, M = ±∞ ). Thus by Theorem 3 R n ( b ) = k ( n +1)! ( b − a ) n +1 for some m ≤ k ≤ M . If one ofequalities holds, then the result is immediate from Theorem 3. Otherwise it follows directly fromDarboux’s intermediate value theorem. (cid:3) Acknowledgments:
We thank L. Polterovich for his advices.
References [1] L. Bers,
On avoiding the mean value theorem , Amer. Math. Monthly (1967) 583.[2] L. W. Cohen, On being mean to the mean value theorem , Amer. Math. Monthly Calculus Third Edition , Cambridge University Press, (2006).
Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
E-mail address : [email protected] School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
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