Online Test Banks
Score higher
See Online Test Banks
Learning anything is easy
Browse Online Courses
Mobile Apps
Learning on the go
Explore Mobile Apps
Dummies Store
Shop for books and more
Start Shopping

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.

  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus

Inside Sweepstakes

Win $500. Easy.