Expressing and Approximating Functions Using the Taylor Series

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:

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:

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:

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.