Taylor’s Series

Definition

Taylor’s Series is a mathematical representation of a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point. It allows us to approximate complex functions using simple polynomials, which is fundamental in numerical methods for solving differential equations and root-finding problems.

Main Content

1. The Taylor Polynomial Concept

• A Taylor series provides a polynomial approximation of a smooth function $f(x)$ near a specific point $a$.
• The approximation becomes more accurate as we include higher-order derivatives (the first, second, third, and so on).

2. General Formula

• The standard Taylor series expansion for a function $f(x)$ centered at $x = a$ is: $f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots$
• When $a = 0$, the series is specifically called the Maclaurin Series.

3. Visual Representation

The graph shows how adding more terms of the Taylor series causes the polynomial curve to hug the original function curve more tightly.

       y |         /--- (Original Function)
         |       /  ... (Taylor Polynomial)
         |     /  .
         |    /  .
         |   / .
         |  / .
---------|------------ x
         a

Working / Process

1. Identify Function and Center Point

• Determine the function $f(x)$ you wish to approximate and the point $x = a$ around which you are centering the expansion.
• Ensure the function is "infinitely differentiable" at that point.

2. Compute Derivatives

• Find the first, second, and third derivatives of the function, $f'(x), f''(x), f'''(x)$, etc.
• Evaluate each derivative at the center point $a$. For example, calculate $f(a), f'(a), f''(a)$, and so on.

3. Construct the Polynomial

• Plug the values obtained into the Taylor formula.
• Simplify the expression to get the final polynomial.
• Example: If approximating $f(x) = e^x$ at $a = 0$ (Maclaurin), the derivatives are all 1, leading to $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$

Advantages / Applications

• Used in numerical analysis to solve ordinary differential equations (Taylor series method).
• Essential for computing values of transcendental functions like $\sin(x)$, $\cos(x)$, and $e^x$ on calculators and computers.
• Enables physicists and engineers to linearize complex non-linear models for easier computation.

Summary

• Taylor’s Series expresses functions as infinite polynomials using derivatives.
• The accuracy of the approximation improves by increasing the number of terms used.
• The Maclaurin series is a special case of Taylor's series where the expansion is centered at zero.
• Important terms: Derivatives, Center point (a), Factorial, Polynomial approximation.