You may have noticed that I left you hanging with the arctangent series a few weeks ago. I told you how to get the series for arctan for x > 1, but not for x < 1. By now, I bet my EM grader has noticed as well, and I've lost a few points. So, a little after the fact, here's how to get a series for arctangent for x < 1.

The useful bit you need that I didn't have is that the arctan of x can also be expressed as (picture 1):

The following picture, (picture 2), has the geometric interpretation that helped me see things more easily as well as the required series. If you think of the tangent of an angle being equal to the opposite side divided by adjacent side, and then take a look at the lower left corner of the picture below, I think you'll see why the expression above is true. It took forever for me to see it after my professor derived it analytically. I think I could have seen it much more quickly derived geometrically like this.

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