Section 1.   Why Taylor Series?

Without attempting to delve too far into this question, we give a few examples here of problems that lend themselves nicely to the use of a Taylor series expansion.

• Consider the need to approximate the value of a transcendental function, such as or . Suppose, for example, you want to find where the argument 3 is in radians. Your first thought would be to simply determine this with the aid of your nearest calculating machine (a calculator, a computer algebra system, etc.). (By the way, how does your calculator determine ?)

  One of the simplest ways to estimate such a value is to find the first few terms of the Taylor series expansion of and use this polynomial P(x) rather than . In essence, assuming P(x) is a good approximation of  , we can just determine P(3), which can easily be done without a calculator, and use this value for  .

• If one wishes to graph an "ugly" function f(x) in a certain interval, it is sometimes easier to determine a Taylor series expansion of f, truncate this to a polynomial P(x), and plot this polynomial. Again, the underlying assumption is that in this interval.

• Suppose you are asked to estimate the value of the following integral:

  

  Such an integral occurs, for example, when one studies the normal distribution in probability and statistics. How do you calculate this? One option is to use the Trapezoidal Rule or Simpson's Rule, which are time-proven techniques. Another is to determine a Taylor polynomial P(x) of the function and to place this P in the integral as the integrand in question. Since you are now simply looking at a polynomial, the integration is easy and the evaluation at the limits of integration should go very smoothly.

  This issue of difficult or impossible integration can also appear during the solution of a differential equation. Hence, it is sometimes the case that solutions will be found in terms of Taylor series as the integration is deemed "performable" in this context.

• As you begin to deal with modeling examples in engineering (mechanical, electrical, etc.), the models in question will often be difficult to handle mathematically as they are. In the literature you can find examples in which the author will then modify the model by studying the "linear approximation" of the model. (This allows for easier integration, solution of a differential equation, or some such thing.) This linear approximation, as one looks closely, is simply the Taylor polynomial of degree 1 for the function in question.