Calculistic is not really a word, as it has yet to make its way into Urban Dictionary, but that's not going to stop me using it to describe the type of manipulation on power series involving (or related to) calculus. But first, let's learn to take the derivative and antiderivative of an infinitely long polynomial. It'll take less than 20 minutes. Duration: 18:47

What function is the same as its derivative and its antiderivative (with the exception of some constant of integration)? Not that the answer has anything at all to do with Example 8. I was just wondering. Duration: 12:10

Example 9 gives us more evidence to suggest that the radius of convergence does not change upon differentiating or integrating a power series. However, the interval of convergence may differ a bit at the endpoints, so check them. Duration: 12:41

Now that we have a fickle grasp on how to differentiate and integrate a power series, let's use that technique to derive a power series expansion for ln(1+x). This one will probably end up on a memory quiz. Duration: 17:25

Now it's your turn. Using the technique modeled in the previous video, generate a power series for arctan(x). To accomplish this task, begin by taking the derivative. That should make an expression that you can manipulate into a power series. Then just integrate your answer to go back to arctan(x). You might want to use arctan(0)=0 as your initial condition, and don't forget to find its interval of convergence. To check your answer, just take a peek into this lesson's slide show.