# Expressing and Approximating Functions Using the Taylor Series

It’s important to understand the difference between *e**xpressing *a function as an infinite series and *a**pproximating *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:

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,* *T** _{n}*(

*x*). Like other polynomials, a Taylor polynomial is identified by its degree. For example, here’s the fifth-degree Taylor polynomial,

*T*

_{5}(

*x*), that approximates e

*:*

^{x}Generally speaking, a higher-degree polynomial results in a better approximation. For the value of *e** ^{x}* when

*x*is near 100, you get a good estimate by using a Taylor polynomial for

*e*

*with*

^{x}*a*= 100:

To sum up, remember the following:

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

A Taylor polynomial,

*T*(_{n}*x*), from a convergent series approximates the value of a function.