Taylor-made Pi

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

Start with

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

/4 = 1 – 1/3 + 1/5 – 1/7 + …

Voila!

There are other  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 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.

How to Cite this Page: 
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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