Review of Taylor Series

Definition:

A Taylor series is an infinite sum of terms that represents a function as a polynomial expression. It is centered at a specific point and expresses the function as a sum of its derivatives.

Formula:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

where:

• f(x) is the function being approximated
• a is the center point of the series
• f'(a), f''(a), f'''(a), ... are the derivatives of f(x) evaluated at a
• n! is the factorial of n (the product of all positive integers up to n)

Convergence:

Taylor series converge for some range of values of x around a. The radius of convergence, R, is the distance from a to the nearest point where the series does not converge.

Cauchy's Integral Formula:

For a function f(z) that is complex-differentiable in an open disc, the Taylor series can be expressed using Cauchy's integral formula:

f(a+z) = 1/(2πi) ∫_γ f(w) / (w-(a+z)) dw

where γ is a circle centered at a with radius r such that |z| < r.

Applications:

Taylor series are widely used in mathematics and physics to:

• Approximate functions by polynomials (e.g., in numerical analysis)
• Solve differential equations (e.g., in celestial mechanics)
• Calculate integrals and derivatives
• Expand functions in more manageable forms (e.g., in Fourier analysis)

Notable Taylor Series Expansions:

• Exponential function: e^x = 1 + x + x^2/2! + x^3/3! + ...
• Sine function: sin(x) = x - x^3/3! + x^5/5! + ...
• Cosine function: cos(x) = 1 - x^2/2! + x^4/4! + ...
• Logarithmic function: ln(1+x) = x - x^2/2 + x^3/3 - ...

Limitations:

Taylor series do not always converge. For example, the Taylor series for f(x) = e^(-x^2) diverges for all x ≠ 0.