Expressing and Approximating Functions Using the Taylor Series

By Mark Zegarelli

It’s important to understand the difference between expressing a function as an infinite series and approximating a function by using a finite number of terms of series. You can think of a power series as a polynomial with infinitely many terms (Taylor polynomial).

Every Taylor series provides the exact value of a function for all values of x where that series converges. That is, for any value of x on its interval of convergence, a Taylor series converges to f(x).

Here’s the Taylor series in all its glory:

A function expressed in terms of the Taylor series.

In practice, however, adding up an infinite number of terms simply isn’t possible. Nevertheless, you can approximate the value of f(x) by adding a finite number from the appropriate Taylor series.

An expression built from a finite number of terms of a Taylor series is called a Taylor polynomial, Tn(x). Like other polynomials, a Taylor polynomial is identified by its degree. For example, here’s the fifth-degree Taylor polynomial, T5(x), that approximates ex:

The fifth-degree Taylor polynomial that approximates ex

Generally speaking, a higher-degree polynomial results in a better approximation. For the value of ex when x is near 100, you get a good estimate by using a Taylor polynomial for ex with a = 100:

using a Taylor polynomial when x equals 100

To sum up, remember the following:

  • A convergent Taylor series expresses the exact value of a function.

  • A Taylor polynomial, Tn(x), from a convergent series approximates the value of a function.