Math 203 Lecture 35 - Power Series and representing functions as power series

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In this lecture we talk about a VERY important kind of series. The power series. This is essentially an infinite polynomial. Polynomials are the nicest functions to do calculus with, and as we see here, this is true for infinite polynomials as well (infinite polynomials = polynomials having an infinite number of terms and no highest degree).

We talk about where these power series converge (absolutely) and then use our knowledge of geometric series to express functions as power series. This is a very powerful technique, that allows us to do calculus, such as compute integrals, for functions that we could not have dealt with before.

The idea of writing functions as absolutely convergent power series is very useful and convenient and we wonder if we can do this for any function. We can't. But there are a LOT of functions for which we can do this.

In the next class, we explore the idea of a Taylor series, and how they can be used to represent almost any function (the infinitely differentiable ones) as power series. As we did in this lecture, we will also explore the consequences and usefulness of this manipulation.