A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. The general formula expresses f(x) as a sum of terms involving the function and its derivatives evaluated at point a, multiplied by powers of (x-a). Here we see the exponential function e to the x and its Taylor approximations centered at x equals 0. As we include more terms, the approximation becomes more accurate, especially near the center point.
Let's derive the Taylor series for sine of x centered at zero, also known as the Maclaurin series. First, we find the derivatives of sine x: sine x, cosine x, negative sine x, negative cosine x, and the pattern repeats. When we evaluate these at x equals zero, we get zero, one, zero, negative one, and so on. Substituting these values into the Taylor series formula, we get x minus x cubed over 3 factorial plus x to the fifth over 5 factorial, and so on. Notice how the Taylor approximations get closer to the sine function as we include more terms, especially near x equals zero.
Let's explore the convergence of Taylor series. The radius of convergence defines the range of x-values for which the series converges to the function. For example, the Taylor series for e to the x, sine x, and cosine x converge for all real values of x. However, for functions like 1 over 1 minus x and natural log of 1 plus x, the series only converge when the absolute value of x is less than 1. In our graph, we can see the function 1 over 1 minus x and its Taylor approximations. Notice how the approximations get closer to the function inside the convergence radius, but diverge outside it. The error in a Taylor approximation can be estimated using the formula shown, which depends on the maximum value of the next derivative and the distance from the center point.
Taylor series have numerous practical applications. First, they're used to approximate functions, allowing calculators and computers to compute values of complex functions like sine, cosine, and exponentials. Second, they help solve differential equations when analytical solutions aren't available. Third, they're essential in physics and engineering fields such as quantum mechanics, signal processing, and control systems. Fourth, they enable error analysis in numerical computations. In our visualization, we can see how the Taylor approximations for cosine x get increasingly accurate as we add more terms. The yellow line shows the error between the actual function and the second-order approximation. Notice how the error increases as we move away from the center point at x equals zero.
To summarize what we've learned about Taylor series: They represent functions as infinite sums of terms calculated from derivatives at a specific point. The general formula expresses f(x) as the sum of f^(n)(a) divided by n factorial, multiplied by (x-a) to the power n. Common examples include the Taylor series for e to the x, sine x, cosine x, and one over one minus x. Each series has its own radius of convergence, which determines where the series accurately represents the function. Taylor series have numerous practical applications in mathematics, physics, engineering, and computer science, including function approximation, solving differential equations, and error analysis in numerical computations.