Taylor's Expansion Revisited: A General Formula for the Remainder

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International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 645736,5pages

doi:10.1155/2012/645736

Research Article

Taylor’s Expansion Revisited: A General

Formula for the Remainder

Jos ´e Juan Rodr´ıguez Cano and Enrique de Amo

Department of Algebra and Mathematical Analysis, University of Almer´ıa, Almer´ıa, 04120 Andaluc´ıa, Spain

Correspondence should be addressed to Enrique de Amo,[email protected]

Received 22 March 2012; Accepted 31 May 2012

Academic Editor: Harvinder S. Sidhu

Copyrightq2012 J. J. Rodr´ıguez Cano and E. de Amo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give a new approach to Taylor’s remainder formula, via a generalization of Cauchy’s generalized mean value theorem, which allows us to include the well-known Sch ¨olomilch, Lebesgue, Cauchy, and the Euler classic types, as particular cases.

1. Introduction

Taylor’s polynomial is a central tool in any elementary course in mathematical analysis. Nowadays, its importance is centred on its applications, for instance, to asymptotic analysis or to obtain satisfactory numerical or integral inequalitiessee, e.g.,1–5. The core of these results comes from manipulations on the explicit formula of the remainder, that is, the error estimation when considering the Taylor’s polynomial expansion instead of the function.

In this paper, we provide a new explicit formula for the remainder that generalizes classic ones, namely, Sch ¨olomilch, Lebesgue, Cauchy, and Euler’s remainders.

Inspired by the explicit expression for an arbitrary polynomial

x−→px, ∀x∈R, 1.1

B. Taylor1712used to write

px

n

k0

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whereais a real parameter and the coeﬃcientsαkare given by their derivatives ofk-order

αk:

pka

k! , k0,1, . . . , n. 1.3

Therefore, in a heuristic way, he introduced, three centuries ago, the expression

fx:

∞

k0

fka

k! x−a

k _1.4

for an arbitrary functionf. Later, but during the same century, A. L. Cauchy gave the name of analytic to a type of functions which stands for their series expansions.It is well known that Cauchy worked to introduce the concept of convergence of series.

An obvious problem is to calculate the formula for the remainder in an explicit form

not only to know that there exists a polynomialpsuch that

fx−px o|x−a|n ifx−→a, 1.5

that is, the functionfhas a contact of order greater thatnin a neighborhood of a pointa. It is also well known that this explicit expression provides an upper bound for the error when we considerpinstead offnear toa.

After Taylor, authors such as Euler, Lagrange, Cauchy, or Schol ¨omilch have considered functions satisfying

fx

n

k0

fka

k! x−a

k_R_x_,

1.6

whereRxmeasures the error when it is not possible to representfin an analytic forme.g., whenfhas derivatives up to then-order, but no further. Ifa0, then the above expression is called the McLaurin series off.

In this paper, we are interested in Taylor’s polynomials to obtain a new and more general explicit form for the remainderRx.

In Section 3.1, via slight modifications on Cauchy’s general mean value theorem

CGMVTfor functionsf with continuous derivatives of ordernona, b, we obtain a very general expression for each of the corresponding classic expressions for the remainder as particular casesSection 3.2.

2. Notation and Preliminaries

Throughout the paper,Rdenotes the set of real numbers andNthe set of the positive integers,

a, bis a closed and bounded interval with endpointsaandb,Cn_{a, b}_{denotes the class}

of all real functions with continuous derivative ofn-order defined ona, b, and the extreme cases:C0_{a, b}_:_C_{a, b} _{the class of all continuous functions defined on}_{a, b}_and

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For givenf∈ Cn_{a, b}_and_x0_∈_{a, b}_{, we denote by}

pf,x0,nx:

n

k0

fkx0

k! x−x0

k_, _∀x_∈_{a, b} _2.2

for Taylor’s polynomial ofn-order centred atx0of the functionf. For the sake of simplicity, we refer to it asp. A functionfis said to be analytic atx0∈a, bif there existsδ >0 such that

fx:

∞

k0

fkx0

k! x−x0

k_, _∀x_∈_x

0−δ, x0δ. 2.3

3. A General Formula for the Remainder

3.1. A General Taylor’s Theorem

The classic technique for obtaining Taylor’s polynomial with a remainder that consists of applying a more general result than the CGMVT is widely known.

Proposition 3.1 n-CGMVT. Let f, g ∈ Cn_{a, b} _{such that} _fn1 _and _gn1 _{exist and are}

continuous on the open intervala, b. Then, there existsξ∈a, bsuch that

fb−

n

k0

fka

k! b−a

k

gn1ξ fn1ξ

gb−

n

k0

gka

k! b−a

k

. 3.1

Note that with this notation, the CGMVT is the corresponding 0-CGMVT. Now, a slight modification in the hypothesis of the above proposition allows a more general statement.

Lemma 3.2n-m-GMVCT. Letf ∈ Cn_{a, b}_and_g _{∈ C}m_{a, b}_{. If}_fn1_{, g}m1 _{∈ C}_{a, b}_,

then there existsξ∈a, bsuch that

fb−

n

k0

fka

k! b−a

k

gm1ξ fn1ξ

gb−

m

k0

gka

k! b−a

k

. 3.2

Proof. We consider auxiliary functions

Fx:

n

k0

fkx

k! b−x

k_,

Gx:

m

k0

gkx

k! b−x

k_, _∀x_∈_{a, b}_.

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They satisfy the CGMVT for functions in C1_{a, b}_{. Moreover, we have the following}

identities:

Fb fb, Gb gb,

Fx f

n1_x

n! b−x

n_, _G_x gm1x

m! b−x

m_.

3.4

Therefore, the result immediately follows.

Theorem 3.3of Taylor. Letf ∈ Cn_{a, b}_and _h _{∈ C}_{a, b}_{, having no zeros in}_{a, b} _(i.e., hx/0 for allxina, b). Suppose that there existsfn1∈ Ca, b. Then, there isξ∈a, bsuch that

fx

n

k0

fka

k! x−a

kfn1ξ

hξ

x−ξn

x−ξm

m!

n!

x

a

x−sm

m! hsds. ∗

Proof. Putg :a, b → Rsatisfying the following conditions:

gm1x:hx ifa≤x≤b,

gka 0 if 0≤k≤m.

3.5

Now, using then-m-CGMVT, we obtain∗.

3.2. Particular Cases

In this subsection, we show how the remainder formula∗can be reduced to each particular case. Firstly, if we define the function h with a constant real value, namely α ∈ R, then Sch ¨olomilch’s version for the remainder followssee6:

Rx fn1ξx−ξ

n

x−ξm

m!

n!

x−am1

m1! . S

This formula is usually obtained directly from the n-CGMVT doing gx : x−an1, becausegka 0, 0≤k≤n, andgn1x n1!.

WhennminS, this formula gives the Lagrange remainder typesee6–9:

Rx f

n1_ξ

n1!x−a

n1_. _L

Using newlyS, if we dom0, the Cauchy formula appears for the remaindersee10:

Rx f

n1_ξ

n! x−ξ

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Now, we return to∗, and puttinghx: fn1xandn m, we obtain Euler’s integral expressionsee11:

x

a

x−sn

n! f

n1_s_ds. _E

Of course, doinga0 in∗, we have a general McLaurin type series:

fx

n

k0

fk0

k! x

kfn1ξ

hξ

x−ξn

x−ξm

m!

n!

x

0

x−sm

m! hsds. ∗ −McL

References

1 M. Akkouchi, “Improvements of some integral inequalities of H. Gauchman involving Taylor’s remainder,” Divulgaciones Matem´aticas, vol. 11, no. 2, pp. 115–120, 2003.

2 H. Gauchman, “Some integral inequalities involving Taylor’s remainder. I,” Journal of Inequalities in

Pure and Applied Mathematics, vol. 4, no. 1, Article ID 1, 5 pages, 2003.

3 H. Gauchman, “Some integral inequalities involving Taylor’s remainder. II,” Journal of Inequalities in

Pure and Applied Mathematics, vol. 3, no. 2, Article ID 26, 9 pages, 2002.

4 Z. Liu, “Note on inequalities involving integral Taylor’s remainder,” Journal of Inequalities in Pure and

Applied Mathematics, vol. 6, no. 3, Article ID 72, 6 pages, 2005.

5 M. Neher, “Improved validated bounds for Taylor coeﬃcient and for Taylor remainder series,” Journal

of Computational and Applied Mathematics, vol. 152, pp. 393–404, 2003.

6 J. Rey Pastor, P. Pi Calleja, and C. A. Trejo, An´alisis Matem´atico, Tomo I. Kapelusz, Buenos Aires, Argentina, 1952.

7 T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Mass, USA, 1974.

8 K. Kuratowski, Introducción al Cálculo, Limusa-Wiley, Mexico City, México, 1970.

9 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Co., New York, NY, USA, 1964.

10 J. M. Ortega, Introducción al Análisis Matemático, Labor Universitaria, Barcelona, Spain, 1993.

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Taylor's Expansion Revisited: A General Formula for the Remainder

Research Article

Taylor’s Expansion Revisited: A General

Formula for the Remainder

Jos ´e Juan Rodr´ıguez Cano and Enrique de Amo

1. Introduction

2. Notation and Preliminaries

3. A General Formula for the Remainder

3.1. A General Taylor’s Theorem

3.2. Particular Cases

References

Submit your manuscripts at

http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific

World Journal

Algebra

Decision Sciences

Discrete Mathematics

Stochastic Analysis