SERIES EXPANSION OF FUNCTIONS, MACLAURIN’S SERIES, TAYLOR’S SERIES, TAYLOR’S FORMULA

Many functions can be represented by polynomials. In this connection let us note a relationshipbetween the coefficients c₀, c₁, c₂, ... ,c_n of the polynomial of degree n

1)        f(x) = c₀ + c₁x + c₂x² + ... + c_nxⁿ

and its derivatives of order one through n at the point x = 0. Let us take the first n derivatives ofpolynomial 1):

            f ‘(x) = c₁ + 2c₂x + 3c₃x² + 4c₄x³ + ...

            f ‘‘ (x) = 2c₂ + 2?3c₃x + 3?4c₄x² + ...

            f ‘‘‘ (x) = 2?3c₃ + 2?3?4c₄x + ... 

            ................................................

            f⁽ⁿ⁾(x) = n!c_n

Putting x = 0 in the above equations and solving for the c’s we obtain

Thus we see that polynomial 1) can be written as

This representation of a polynomial provides us with a means of representing more or lessgeneral functions by polynomials.

There is a completely analogous relationship between the coefficients c₀, c₁, c₂, ... ,c_n of thepower series in x-a of degree n

3)        f(x) = c₀ + c₁(x-a) + c₂(x-a)² + ... + c_n(x-a)ⁿ

and its derivatives of order one through n at the point x = a. If we take the first n derivatives ofpower series 3) we get:

            f ‘(x) = c₁ + 2c₂(x-a) + 3c₃(x-a)² + 4c₄(x-a)³ + ...

            f ‘‘ (x) = 2c₂ + 2?3c₃(x-a) + 3?4c₄(x-a)² + ...

            f ‘‘‘ (x) = 2?3c₃ + 2?3?4c₄(x-a) + ... 

            ................................................

            f⁽ⁿ⁾(x) = n!c_n

Putting x = a in the above equations and solving for the c’s we obtain

Thus we see that power series 3) can be written as

Since a power series is a polynomial, this is another polynomial representation of a function. Note that 2) is a special case of 4) where a = 0. Also note that if we allow n to increase withoutlimit 2) and 4) become infinite series. An infinite series of type 2) is called Maclaurin’s seriesand an infinite series of type 4) is called Taylor’s series. Such series may be used to representrather general functions within some interval of convergence.

When some function f(x) is written in the form of an infinite series, the function is said to beexpanded in an infinite series and the infinite series is said to represent the function in theinterval of convergence.

Maclaurin’s Series. A series of the form

Such a series is also referred to as the expansion (or development) of the function f(x) in powersof x, or its expansion in the neighborhood of zero. Maclaurin’s series is best suited for findingthe value of f(x) for a value of x in the neighborhood of zero. For values of x close to zero thesuccessive terms in the expansion grow small rapidly and the value of f(x) can often beapproximated by summing only the first few terms.

A function can be represented by a Maclaurin series only if the function and all its derivativesexist for x = 0. Examples of functions that cannot be represented by a Maclaurin series: 1/x, ln x,cot x.

Example 1. Expand e^x in a Maclaurin Series and determine the interval of convergence.

Solution. f(x) = e^x, f ‘(x) = e^x, f ‘‘(x) = e^x, f ‘‘‘(x) = e^x, ........ , f⁽ⁿ⁾(x) = e^x

and

            f(0) = 1, f ‘(0) = 1, f ‘‘(0) = 1, f ‘‘‘(0) = 1, ....... ,f⁽ⁿ⁾(0) = 1

so

Example 2. Expand sin x in a Maclaurin Series and determine the interval of convergence.

Solution. f(x) = sin x, f‘(x) = cos x, f‘‘(x) = - sin x, f‘‘‘(x) =- cos x, ......

Since sin 0 = 0 and cos 0 = 1 the expansion is

It converges for all values of x since

Taylor’s Series. A series of the form

This series is useful for computing the value of some general function f(x) for values of x near a.. The nearer to a the value is, the more quickly the series will converge. When a = 0, Taylor’sSeries reduces, as a special case, to Maclaurin’s Series. If we set x = a + h, another useful formof Taylor’s Series is obtained:

In terms analogous to those describing Maclaurin’s expansion, Taylor’s series is called thedevelopment of f(x) in powers of x - a (or h), or its expansion in the neighborhood of a.

Taylor’s Formula with the Remainder. Let a function f(x) and its first n+1derivatives be continuous on a closed interval containing x = a. Then there is a number x₀between a and x such that

in which R_n+1(x), the remainder after n + 1 terms, is given by the formula

The remainder has been put in several different forms, the usefulness of the form depending onthe particular type of function being expanded. The above form is Lagrange’s form. Thetheorem with Lagrange’s form of the remainder is a simple variation of the Extended Law of theMean.

Taylor’s theorem is one of the most important theorems in applied analysis. It describesapproximating polynomials for a broad range of functions and provides estimates for errors. Many of the laws of nature, physical and chemical processes, the motion of bodies, etc. areexpressed with great accuracy by functions which can be expanded in a Taylor’s series.

Theorem 1. The Taylor Series expansion of a function f(x) is a valid representation of thefunction for those values of x, and only those values, for which the remainder R_n+1(x) approacheszero as n becomes infinite.

Maclaurin’s Formula with the Remainder. Let a function f(x) and its first n+1derivatives be continuous on a closed interval containing x = 0. Then there is a number x₀between 0 and x such that

in which R_n+1(x), the remainder after n + 1 terms, is given by the formula

This is a special case of Taylor’s Formula, obtained by letting a = 0.

Theorem 2. The Maclaurin Series expansion of a function f(x) is a valid representation of thefunction for those values of x, and only those values, for which the remainder R_n+1(x) approacheszero as n becomes infinite.

References.

  Mathematics, Its Content, Methods and Meaning. Vol I

  James and James. Mathematics Dictionary.

  Middlemiss. Differential and Integral Calculus.

  Oakley. The Calculus. p. 190 - 193

  Ayres. Calculus. p. 242, 248

  Smith, Salkover, Justice. Calculus. p. 372 - 383

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