Post 3: The Taylor Series Expansion

In calculus, the Taylor series expansion is a way to approximate a function using an infinite series of terms derived from its derivatives. It is named after the mathematician Brook Taylor, who first introduced this concept in the 18th century. The Taylor series expansion is a powerful tool that allows us to represent a wide range of functions in terms of simple polynomial expressions.

Definition: The Taylor series expansion of a function f(x) centered at the point a is given by:

where n! denotes the factorial of n and f⁽ⁿ⁾(a) represents the n-th derivative of f evaluated at a.

Conditions for a Valid Taylor Series: For a function to have a valid Taylor series representation, it must satisfy certain conditions. These conditions include the function being infinitely differentiable in a neighborhood of the expansion point a, and the existence of higher-order derivatives at every point in that neighborhood.

Example: Let's find the Taylor series expansion of the function f(x) = sin(x) centered at a = 0.

Step 1: Calculate the derivatives of f(x):

• f'(x) = cos(x)
• f''(x) = -sin(x)
• f'''(x) = -cos(x)

Step 2: Evaluate the derivatives at a = 0:

• f(0) = sin(0) = 0
• f'(0) = cos(0) = 1
• f''(0) = -sin(0) = 0
• f'''(0) = -cos(0) = -1

Step 3: Substitute these values into the Taylor series expansion formula: f(x) ≈ 0 + (1/1!)(x - 0) + (0/2!)(x - 0)² + (-1/3!)(x - 0)³ + ...

Simplifying the above expression, we get: f(x) ≈ x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ...

This is the Taylor series expansion of f(x) = sin(x) centered at a = 0.

The Taylor series expansion provides us with a way to approximate the values of the sine function for a given range of x values. By including more terms in the series, we can achieve a more accurate approximation.