Interpreting the power series Σx^n geometrically, we should be able to find the sum of the series for every value of x within its interval of convergence. If we plot these points in the coordinate plane, we get the graph of a continuous and differentiable function. It is this function to which our power series converges. Duration: 10:11

Now let's reverse the process used in the Warm-up to find a function to which a power series converges. Here, we want to start with the function, then find its power series expansion. We'll begin geometrically. Duration: 15:35

The next few theorems give us a variety of ways to generate a power series for a function through algebraic manipulation. Since a power series is basically an infinitely long polynomial, we can do things like function composition to create a new series. Duration: 9:41

Think of the Distributive Property. Now imagine performing that property an infinite number of times. That's what we are about to do. It may take a while. Duration: 9:01

As Example 5 demonstrations, we can also create a new series by adding or subtracting the terms of two other series. Duration: 5:56