Section Summary

In this section we have formalized the observations of the preceding section about values of polynomial approximations approaching values of non-polynomial functions as the degree of the approximation is increased. We have introduced the language of infinite series — in particular, the concept of Taylor polynomial leads in a natural way to Taylor series as a “polynomial of infinite degree.” This is a theme that we will develop further in subsequent sections.

We find that the question of convergence of a sum with infinitely many terms is intimately related to convergence of a sequence, namely, the sequence of partial sums. The idea of sequence convergence has now appeared several times — e.g., in Euler's Method, Newton's Method, and in approximating sums for definite integrals.

As a side benefit of having introduced the Taylor series for the exponential function, we have now been able to show that this function grows faster than any power function.

Since any arithmetic machine can evaluate a polynomial expression, this could be how your calculator or computer finds very accurate values of non-polynomial functions. The actual routines programmed in these devices are somewhat more complicated — for greater efficiency — but the idea is the same. We illustrated this by showing how to calculate values of the exponential, sine, and error functions via simple arithmetic.