Taylor Series | Engineering Mathematics - Engineering Mathematics PDF Download

Taylor series is the representation of a function as an infinite sum of terms, each term being a constant multiplied by a power of (x - a). Each successive term has a higher power (exponent) than the previous term. The Taylor series expansion is centred at a point a. When the centre a = 0, the series is called the Maclaurin series.

Taylor's Series Theorem

The Taylor series of a function f(x) that is infinitely differentiable at a point a is the power series

In sigma notation this is written as

Here f⁽ⁿ⁾(a) denotes the n^th derivative of f evaluated at x = a, and n! is the factorial of n. This representation gives a polynomial approximation of increasing degree; under suitable conditions the infinite sum converges to the value f(x).

Explanation of terms and convergence

Coefficients: The coefficient of (x - a)ⁿ is f⁽ⁿ⁾(a)/n!. Convergence: The radius of convergence of the Taylor series depends on the nearest singularity of f in the complex plane; inside that radius the series converges absolutely. If the series converges to f(x) for all x in some interval containing a, f is called analytic at a.

Remainder (Error) term

To measure the difference between f(x) and the n-th degree Taylor polynomial T_n(x), define the remainder R_n(x) by

f(x) = T_n(x) + R_n(x).

One common form of the remainder (Lagrange form) is

R_n(x) = f⁽ⁿ⁺¹⁾(ξ)/(n+1)! · (x - a)ⁿ⁺¹ for some ξ between a and x.

This form is useful to bound the error when the (n+1)^th derivative is known or can be bounded on the interval between a and x.

Proof (derivation of coefficients)

Begin by assuming f can be expanded as a power series about x = 0 (Maclaurin form):

Differentiate term-by-term.

Evaluate the series and its derivatives at x = 0 to obtain the coefficients.

For example, f(0) = a₀. f′(0) = a₁. f″(0) = 2 a₂ ⇒ a₂ = f″(0)/2!. By induction, a_n = f⁽ⁿ⁾(0)/n!.

Substituting these coefficients into the power series gives the Maclaurin series:

To generalise about a point a, replace x by (x - a) and proceed similarly. Write

f(x) = b + b₁(x - a) + b₂(x - a)² + b₃(x - a)³ + ...

Evaluating derivatives at x = a gives b_n = f⁽ⁿ⁾(a)/n!. Substituting yields the Taylor series about a:

Taylor series of sin x (Maclaurin at x = 0)

Derivatives cycle every four steps:

f(x) = sin x

f′(x) = cos x

f″(x) = - sin x

f‴(x) = - cos x

f⁽⁴⁾(x) = sin x, and so on.

Evaluating at x = 0 and substituting into the Maclaurin formula gives only odd powers of x with alternating signs:

The Maclaurin series for sin x converges for all real x (radius of convergence ∞).

Taylor series in several variables

Functions of several variables also admit Taylor expansions when sufficiently differentiable. For a function f(x, y) expanded about (a, b), the second-order form is

The full multivariable expansion contains mixed partial derivatives and terms of higher total degree. Multivariate Taylor polynomials are widely used in optimisation, numerical methods, and approximations in engineering problems.

Maclaurin series (special case a = 0)

If the Taylor series is centred at 0 (a = 0) it is called the Maclaurin series. The general Maclaurin form is

Example: For |x| < />

1/(1 - x) = 1 + x + x² + x³ + x⁴ + ...

This geometric series converges for |x| < 1. many standard maclaurin expansions x, sin x, cos x, ln(1 + x) for |x| < 1, (1 +α by binomial series) are used as building blocks in analysis and approximation.

Applications of Taylor series

• Taylor series reconstruct a function from its derivatives at a single point, giving an analytical local representation.
• Taylor expansions simplify many mathematical proofs by replacing functions with polynomials.
• Partial sums (Taylor polynomials) provide polynomial approximations useful in numerical computation and engineering design.
• Multivariate Taylor series are fundamental in optimisation methods (e.g. Newton's method, quadratic approximations).
• Taylor series appear in electrical engineering applications such as linearisation around operating points and in power-flow analysis of electrical power systems.

Problems and Solutions

Question 1: Determine the Taylor series at x=0 for f(x) = e^x

Soln:

Write the function: f(x) = e^x.

Differentiate repeatedly: f′(x) = e^x; f″(x) = e^x; f‴(x) = e^x; and so on.

Evaluate derivatives at x = 0: f(0) = 1; f′(0) = 1; f″(0) = 1; f‴(0) = 1; ...

Substitute into the Maclaurin formula f(x) = Σ f⁽ⁿ⁾(0)/n! · xⁿ to get the series.

Therefore,

e^x = 1 + x + x²/2! + x³/3! + ..., for all real x.

Question 2: EValuate the Taylor Series for f ( x ) = cos ( x ) for x = 0.

Soln:

Compute derivatives and evaluate at 0:

f(x) = cos x ⇒ f(0) = 1

f′(x) = - sin x ⇒ f′(0) = 0

f″(x) = - cos x ⇒ f″(0) = -1

f‴(x) = sin x ⇒ f‴(0) = 0

f⁽⁴⁾(x) = cos x ⇒ f⁽⁴⁾(0) = 1

The derivatives repeat every four steps.

Substitute into the Maclaurin series formula.

Hence the Maclaurin series is

cos x = 1 - x²/2! + x⁴/4! - x⁶/6! + ..., valid for all real x.

Question 3: Evaluate the Taylor Series for f (x) = x³ - 10x² + 6 at x = 3.

Soln:

Compute the function value and derivatives at x = 3.

f(x) = x³ - 10x² + 6 ⇒ f(3) = 27 - 90 + 6 = -57

f′(x) = 3x² - 20x ⇒ f′(3) = 27 - 60 = -33

f″(x) = 6x - 20 ⇒ f″(3) = 18 - 20 = -2

f‴(x) = 6 ⇒ f‴(3) = 6

Higher derivatives are zero: f⁽⁴⁾(x) = 0 and beyond.

Form the Taylor series about a = 3 using T_n(x) = Σ f⁽ⁿ⁾(3)/n! · (x - 3)ⁿ.

Thus the series ends at the third derivative (polynomial):

f(x) = f(3) + f′(3)(x - 3) + f″(3)/2!(x - 3)² + f‴(3)/3!(x - 3)³.

Notes and useful remarks

• If f is a polynomial of degree m then the Taylor series about any point a terminates after degree m and equals f(x) for all x.
• Even functions have series with only even powers; odd functions only odd powers.
• Checking the remainder term is essential when using a finite Taylor polynomial to approximate f(x); bound the remainder using a suitable estimate of the (n+1)^th derivative.
• Taylor expansions are widely used in engineering for linearisation about an operating point, error estimation, numerical integration, and solving differential equations approximately.