In the preceding section we saw that

1. some familiar important functions can be approximated by polynomial functions, and
2. the coefficients of these polynomials can be calculated from derivatives of the original functions.

We found that these approximations can be quite accurate close to the reference point `x = 0`, but the accuracy deteriorates as we move away from that point. On the other hand, accuracy appears to improve as the degree of the approximating polynomial increases. This suggests that we may want to consider what happens as the degree becomes quite large — perhaps even to consider limiting values as the degree goes to infinity. That will be the subject of this section.

We note in passing two features of Taylor polynomials that emerge from our study thus far:

• We don't waste anything by computing low-degree polynomials on the way to finding higher-degree polynomials. All the low-degree terms are also terms in the higher-degree approximations. Indeed, each `P_n` can be computed from `P_(n-1)` by adding one more term. And that next term can be calculated by an easily automated process: Take one more derivative, evaluate at `x=0`, divide by `n!`, and multiply by `x^n`.

• On the other hand, this direct calculation of additional terms depends on symbolic calculation of higher-order derivatives. So far we have done that only for functions whose higher-order derivatives are easy to calculate — but most functions aren't like that. Thus, if we are going to find approximations of high degree, we need to be on the lookout for easier ways to find the coefficients.

10.3.1 What Happens as the Degree Increases?

If `P_n text[(] x text[)]` is the `n`th degree Taylor approximation to the function `e^x`, then for each real number `x_0`, the numbers `P_n text[(] x_0 text[)]` approach `e^(x_0)` as the degree `n` increases. Similarly, if `P_n text[(] x text[)]` is the `n`th degree Taylor approximation to the function `sin x`, then for each real number `x_0`, the numbers `P_n text[(] x_0 text[)]` approach `sin x_0` as the degree `n` increases. We illustrate these convergences numerically in Tables 1 and 2 by showing the successive approximations to `e^2 = 7.389056` and `sin text[(] -pi`/`2 text[)] = -1`, respectively.

Now we investigate these convergences graphically.

Activity 1

1. Use your graphing tool to explore the convergence of the Taylor polynomials for `e^x` at a variety of values `x_0`. You will see more clearly what is happening at values `x_0` that are two or more units away from `0`.

2. Make the necessary changes in your worksheet, and explore the convergence of the Taylor polynomials for `sin x` at a variety of values `x_0.` To review the approximating polynomials for `sin x`, see the Comment on Activity 2 in the preceding section.

In these two examples, our values of `x_0` are not terribly far from zero, and the approach to `f text[(] x_0 text[)]` is fairly rapid in each case. For reasons we will see later, the statement about Taylor polynomial values approaching values of the exponential and sine functions is correct even for very large values of `x_0.`