Convergence of Taylor Series

The Taylor series provides a systematic method for approximating a smooth function by a polynomial that closely matches the function near a chosen point. This approach is particularly valuable in scientific and engineering contexts where functions may be difficult to evaluate directly, such as oscillatory voltages in alternating current (AC) circuits. Replacing complex functions with polynomial expressions simplifies computation while preserving essential local behavior. Taylor’s Theorem establishes the theoretical foundation for constructing these approximations and for determining their accuracy.

The Taylor polynomial of order n is formed by matching the function’s value and its derivatives up to order n at a central point, typically denoted by a. Each derivative captures additional local information about the function, including its rate of change and higher-order variations. By incorporating successive derivatives, the polynomial increasingly resembles the original function within a neighborhood of the point a. This systematic inclusion of higher-order terms refines the approximation and improves its local accuracy.

Taylor’s Theorem also introduces a remainder term, which quantifies the difference between the function and its nth-degree Taylor polynomial. The remainder depends on the derivative of order n plus one evaluated at some point between a and the point of interest. As n increases, the magnitude of this remainder near a typically decreases, provided the function is sufficiently smooth. If the remainder approaches zero as n approaches infinity for all values within an interval I, then the Taylor series converges to the original function on that interval. This convergence confirms that the infinite series representation accurately reproduces the function throughout its interval of convergence, ensuring both theoretical validity and practical reliability in applications.