Convergence of Taylor Series

Many scientific problems involve functions that are difficult to compute directly, such as oscillatory voltages in AC circuits.

In practice, these functions are often replaced with polynomial expressions that are easier to evaluate.

Taylor’s Theorem gives a systematic way to build these polynomial approximations. It states that a function near a chosen point, a, equals a Taylor polynomial of order n, plus a remainder term.

The Taylor polynomial is a finite part of a Taylor series, built from the function’s value and derivatives up to order n at a. Each new term captures more of the function’s local behavior, so the approximation often improves near a.

The remainder term measures the approximation error. It is the difference between the actual function value and the polynomial value at the same point. This error depends on the derivative of order n plus one at some point between a and x. As n increases, the error near a often gets smaller.

If the remainder approaches zero as n approaches infinity for all x in an interval I, then the Taylor series converges to the function on that interval.

This means that the series represents the function on that interval.