Applications of Taylor Series

The Taylor series expresses functions as an infinite power series over an interval of convergence around a chosen center.

When the center is zero, the series is called a Maclaurin series, which gives a simpler form using powers of x and derivatives at zero.

A practical example is an ideal mass-spring system moving in simple harmonic motion.

The motion of the mass can be described using a cosine function.

In high-speed computing, repeatedly evaluating the cosine function can be computationally intensive. To simplify this, the cosine function is approximated with its Maclaurin series for small time values near zero.

The first term gives a constant value equal to the starting position. Adding the next term introduces curvature, so the approximation matches the motion more closely. Including higher-order terms improves accuracy and captures more of the oscillation.

For small time intervals near zero, only a few terms can provide a good approximation. As time moves farther from the center, more terms are needed to maintain accuracy.

This is why the Taylor series is valuable: it simplifies complex functions while matching the spring’s motion near the center.