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Expressing Functions as Power Series Using the Taylor Series

The Taylor series provides a template for representing a wide variety of functions as power series. It is relatively simple to work with, and you can tailor it to obtain a good approximation of many functions.

Here’s the Taylor series in all its glory:


The Taylor series uses the notation f(n) to indicate the nth derivative. Here’s the expanded version of the Taylor series:


The presence of the variable a provides the Taylor series with a lot of flexibility, as the next example illustrates.

Suppose you want to approximate the value of sin 10. You can use only four terms of the Taylor series to make a good approximation. The key to this approximation is a shrewd choice for the variable a:

Let a = 3

This choice has two advantages: First, this value of a is close to 10 (the value of x), which makes for a good approximation. Second, it’s an easy value for calculating sines and cosines, so the computation shouldn’t be too difficult.

To start off, substitute 10 for x and 3 for a in the first four terms of the Taylor series:


Next, substitute in the first, second, and third derivatives of the sine function and simplify:


The good news is that sin 3 = 0, so the first and third terms fall out:


At this point, you probably want to grab your calculator:


This approximation is correct to two decimal places.

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