Bringing a 'Series' of Wonders to the Calculus Classroom

The following may drive away most casual readers but it does describe what I try to do every day. One of my goals in starting this blog was to enable a dialogue for effective instructional strategies. My focus has generally been on middle and secondary school curriculum up to Algebra 2, bordering on Precalculus. Today I am sharing a different experience. I hope some of you will appreciate it beyond its technical aspects. Similar developments can be found in some textbooks and experienced teachers already do most of this but as this scenario is fresh in my mind, I thought I'd re-play it for you...

Although most Advanced Placement Calculus (BC) teachers are completing or have already completed the unit on infinite series, I would like to offer a view that I hope brings a sense of 'shock and awe' to the student of the 21st century who rarely has the time to stop and appreciate the beauty of our subject. To those who have been teaching this for a while, you may not quite feel this. However, I still get goosebumps when I observe student reactions as this unfolds in front of their eyes...

Assume that students already have a basic understanding of infinite series, the infinite geometric series in particular.

Consider the following three infinite geometric series:

1+1/2+1/4+1/8+... = 1/(1-1/2) = 2
1+1/3+1/9+1/27+... = 1/(1-1/3) = 3/2
1-1/4+1/16-1/64+... = 1/(1-(-1/4)) = 4/5

Just a collection of simple geometric series, boys and girls?
Genius is looking at an ordinary collection of objects and seeing something different. Some mathematician or mathematicians (research this and report back with their bios!) may have considered a reverse view of these series. Instead of the goal being a formula for the sum of the series, perhaps the goal was to represent a function in a different way. Step back into history...

Consider the general formula for the sum of all these series: 1/(1-r) provided r is between -1 and 1. Replace r by x, the variable we usually use for functions, and we can write:
1 + x + x² + x³ +... = 1/(1-x) provided x is between -1 and 1.

The 'polynomial' of infinite degree on the left is known as a power series in x. As long as x is between -1 and 1 (the interval of convergence), this 'equation' makes sense and allows us to use algebraic and calculus operations to represent other related functions. Think of how one might have felt when 'discovering' this and I'm just speculating here. The rational function 1/(1-x) is being represented by some kind of polynomial that never ends. Even though x= 1 is not in the interval of convergence, substituting leads to 1/0 = 1 + 1 + 1 + 1 +.... Hmm....
Let's try substitution on this representation.
Replace x by -x²:
(You can show the domain is unchanged)
1/(1 - (-x²)) = 1 + (-x²) + (-x²)² + (-x²)³ + ... OR
1/(1 + x²) = 1 - x² + x⁴ - x⁶ +...

Ok, let's integrate both sides (assuming it's legal to do so):
tan^-1(x) = x - (1/3)x³ + (1/5)x⁵ - (1/7)x⁷ + ... + C
Replacing x by 0, we see that C = 0.
Now, you'll have to accept this for the moment (to be proved later), equality holds for x = 1, even though 1 was not in the original interval of convergence! It is not unusual when integrating a power series to see the domain include one or both endpoints even though the original function excluded them!

Thus, tan^-1(1) = 1 - 1/3 + 1/5 - 1/7 + ...
Anyone recognize the left-hand side?
The bell rings...