In my two posts immediately preceding this one I suggested an approach to introducing power series by kind of sneaking up on them starting with the tangent line (local linear) approximation and then going for a second-, third- and higher-degree polynomial that had the same value and same derivative values as the function at a point. These polynomials are called Taylor Approximating Polynomials centered at x = a. If a = 0 then they are called Maclaurin polynomials.
Next, we looked at the graphs of two of these one for ln(x) and one for sin(x). If you tried some others, I suggest you be sure to look at their graphs. We saw that the graphs of the polynomials “hugged” the original function and the higher the degree the closer they came to the function. But there was a difference: the ln(x) polynomials were close only in an interval about two units wide with x = a in the center, while the polynomials for sin(x) seemed to be close to sin(x) over wider and wider intervals.
This brought several questions to mind. Hopefully, you can draw these and other questions out of your class and then discuss them as a preview of coming ideas and motivation to learn more . Here are the questions.
1. If there were an infinite number of terms, would the polynomial (now more properly called an infinite series) be the same as the function? Equal, that is. We are of course used to thinking in terms of limits by now. For some functions, such as the sin(x), it appears that this might be the case. But for others, such as ln(x), certainly not.
2. How do you add an infinite number of terms? Good question to which I don’t have a ready answer. In cases like the sin(x) it looks like the sum would be sin(x).
3. Over what interval is the approximation “good”? How do you find the interval? We need a way to find this interval, called the interval of convergence. The interval, as it turns out, can usually be found using the Ratio Test or some other means, For example, if the polynomial turns out to be a geometric series, then the interval depends on the common ratio of the series. Is the interval the same for all functions? No; look at the two examples we have been working with.
4. How good is the approximation? A question you should ask with any approximation. There are several ways of determining this. The two primary ones are the Alternating Series Error Bound and the Lagrange Error Bound. I will discuss error bounds in a later post.
5. Is there an easier way to build the polynomial? Do you have to figure out and evaluate all of the derivatives? Luckily, no. There are easier ways to find a number of series and that too will be the subject of a later post. But not all series; occasionally you will need to find the derivatives and do all the computations. So some practice with that is in order.
6. So okay, this is a lot of fun, but why bother? Polynomials are really easy to handle. They are easy to evaluate, differentiate and integrate. Other functions, not so much. We all learned that and values for other “special angles”, but what is sin(0.2)? Of course you usually find such information on a calculator: sin(0.2) = 0.198669331, but you can also find it with a few terms of the series for sin(x):
Notice that using only three terms you have an answer correct to 8 decimal places. So one answer is that you can use the approximating polynomials to, wait for it, approximate. But there are other uses as well. Stay tuned.