After learning about the Taylor series for 1/(1+x) in calculus, you can find an interesting expression for Pi very easily.

1/(1+w) = 1 – w + w2 – w3 + …

Now substitute x2 for w:

1/(1+x2) = 1 – x2 + x4 – x6 + …

Then integrate both sides (from x=0 to x=y):

arctan y = y – y3/3 + y5/5 – y7/7 +…

and plug in y=1, to get

Pi/4 = 1 – 1/3 + 1/5 – 1/7 + …

Voila!

There are other pi formulas that converge faster.

Presentation Suggestions:
An alternate way to present this is to start with the well-known formula for Pi, and then present this as a “justification”.

The Math Behind the Fact:
Well, we glossed over the issue of why you can integrate the infinite series term by term, so if you wish to learn about this and more about Taylor series, this material is often covered in a fun course called real analysis.

Su, Francis E., et al. “Taylor-made Pi.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Arthur Benjamin

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