Taylor Series

• Taylor series forms a foundation for approximating functions and solving difficult equations.
• The main fundamental concept of Taylor series is that any function can be represented as an infinite sum of terms computed based on its derivatives at a certain point.
• The general form for a Taylor series can be expressed as: f(x) = f(a) + f’(a)(x-a) + f’‘(a)(x-a)^2/2! + f’’‘(a)(x-a)^3/3! + …, where f(n)(a) represents the nth derivative of the function evaluated at a.
• The remainder of a Taylor series tells us the difference between the function and the polynomial approximation, which shrinks as more terms are added.
• In some special cases, the resulting series from Taylor Series can be an exact representation of the actual function. Examples are the exponential function, sine, cosine, etc.
• Maclaurin series is a special case of the Taylor series, where the expansion is about the point a=0. In this case, the Taylor series simplifies to: f(x) = f(0) + f’(0)x + f’‘(0)x^2/2! + f’’‘(0)x^3/3! + …
• A popular method to estimate the error when we truncate the series is using the Lagrange form of the remainder. This helps in knowing how good the approximation is.
• The Radius of Convergence is the interval of x-values where the series converges.
• The Ratio Test can be used to find the Radius of Convergence.
• Some functions may have certain breakpoints preventing the Taylor series from perfectly approximating them. These are usually caused by discontinuities in the original function or its derivatives.
• Calculating data using Taylor series is pivotal in fields like Physics and Engineering.
• Proficiency in manipulating series and their calculus is necessary for understanding and applying Taylor series.