 Welcome to this quick recap of section 8.6 on power series. We'll begin with a quick reminder from section 8.5 on Taylor series. The Taylor series of a function f centered at x equals a is given by this infinite series below. The parts highlighted in red are often called the coefficients of the series because they're the numerical values in front of the polynomial parts of the series. It's worth taking a close look at these. Each of these red highlighted parts involves a derivative of the function f evaluated at x equals a and a factorial. The key thing to notice is that each of those is a number, and so these red parts, even though they might look complicated, is really just a number in front of a polynomial. By taking that idea, we can define a new thing, which is called a power series. A power series centered at x equals a is a function in this form, and you notice that this form is extremely similar to the form above. In fact, the difference is we've replaced this complicated part in front of each polynomial with something called a c sub k. The sequence c sub k is a sequence of real numbers, and it can represent any list of real numbers that we want. If we'd like, we could make a Taylor series from a power series by making those coefficients c sub k have just the right form using derivatives and factorials. But we could really use any numbers we want. Here, x is a variable, and so we really do have a function. A key thing to notice here is that a power series really is a function. It's a function of the variable x, and it's represented as an infinitely long polynomial. A key difference between Taylor series and power series is that for a Taylor series, we had another function in mind, such as e to the x or sine of x, and we calculated this infinitely long polynomial using the derivatives of that function. For a power series, we don't have any other function in mind. We've made a new function, and this infinitely long series is a function in its own right. Just like a Taylor series, however, a power series has an interval of convergence. This is the domain of this function, and it's important to think of it this way. The interval of convergence is the list of x values for which this function is defined. And just like a Taylor series, it's centered at x equals a, and it has a radius of convergence that we call r. This is an interval, and the endpoints may be open or closed. It may be as small as a single number, which would be just the center a, or it can be as large as the entire real number line from negative infinity to infinity. Again, you should think of this as the interval of x values, where the function defined by this series is defined. So it is its domain. Every Taylor series is a power series. So see the book for some standard series that you should know, listed by what their coefficients are, what the CKs are. We can treat power series just like we can treat any other functions. The reason we care about these is that power series really are infinitely long polynomials, and polynomials are especially easy to work with. Here's an example. This is a Taylor series, but remember that we can make power series that don't have any other simple formula. We know if Taylor series, that is a power series, for e to the x, and here it is. But we don't just have to use it as it is here. We can evaluate this at x equals 1, and that gives us an infinitely long series for the number e. It's easier to calculate the value of this series than it is to try to figure out what e means in any other way. We can do more, however. For example, what if we wanted to know what e to the negative x squared is? We can substitute negative x squared into our formula for e to the x, and by substituting that into the polynomial on the right, we get another polynomial. Make sure you understand where all of the negative x squareds came from in the right here. Simplifying a little, we get another nice-looking polynomial that is another way to represent the function e to the negative x squared. Furthermore, we can take derivatives of polynomials very easily. So taking the derivative of e to the x is the same as taking the derivative of its power series on the right. Take a look at this series, and make sure you understand how we took the derivative. And also notice, it's very nice to take the derivative of a polynomial. They're the first kinds of functions we worked with in calculus, and they're the simplest ones to take the derivative of to evaluate, or to evaluate at a new function. Power series can be differentiated and integrated, especially nicely, as we saw in the last example. We call this method integrating or differentiating term by term. So if we have a power series, like the one written here, we can take its derivative by taking the derivative of each term of this polynomial. This complicated-looking bit in the front in the middle here really is just what we get when we take the derivative of x to the k with a coefficient ck out front. k taken down front, and an x minus k in the exponent. On the right, you can see how we took this derivative term by term in the polynomial. Similarly, we can find the general antiderivative for this series by integrating each term. Again, we have a k plus 1 in the power, dividing by k plus 1, just like we would find a general antiderivative for any other polynomial. And again, on the right, you can see how this looks when we do it to the polynomial term by term. The especially amazing thing about this is that if our original series, f of x, converges from negative r to r, then so do its derivative and antiderivative series. The endpoints at negative r and r should be checked separately because they might change. What this really says is that if you have a power series, then its derivative and antiderivative series have the same interval of convergence, except possibly at the endpoints. Now that we've seen these, let's take a look at how to use these and calculate these in practice.