t’s important to understand the difference between [i]e[/i][i]xpressing [/i]a function as an infinite series and [i]a[/i][i]pproximating [/i]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 [i]x[/i] where that series converges. That is, for any value of [i]x[/i] on its interval of convergence, a Taylor series converges to [i]f[/i]([i]x[/i]).Here’s the Taylor series in all its glory:[br][br][img width=171,height=56]https://www.dummies.com/wp-content/uploads/312184.image0.png[/img][br][br]In practice, however, adding up an infinite number of terms simply isn’t possible. Nevertheless, you can approximate the value of [i]f[/i]([i]x[/i]) 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 [i]Taylor polynomial,[/i] [i]T[/i][i]n[/i]([i]x[/i]). Like other polynomials, a Taylor polynomial is identified by its degree. For example, here’s the fifth-degree Taylor polynomial, [i]T[/i]5([i]x[/i]), that approximates e[i]x[/i]:[br][br][br][img width=204,height=41]https://www.dummies.com/wp-content/uploads/312185.image1.png[/img][br][br]Generally speaking, a higher-degree polynomial results in a better approximation. For the value of [i]e[/i][i]x[/i] when [i]x[/i] is near 100, you get a good estimate by using a Taylor polynomial for ex: with [i]a[/i] = 100:[br][br][img width=525,height=80]https://www.dummies.com/wp-content/uploads/312186.image2.png[/img][br][br][br]To sum up, remember the following:[list][*]A convergent Taylor series expresses the exact value of a function.[/*][*]A Taylor polynomial, [i]T[/i][i]n[/i]([i]x[/i]), from a convergent series approximates the value of a function.[/*][/list]