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From the Fun Fact files, here is a Fun Fact at the Advanced level:

Taylor-made Pi

Figure 1
Figure 1

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

Start with

1/(1+w) = 1 - w + w2 - w3 + ...
Now subsitute 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.

How to Cite this Page:
Su, Francis E., et al. "Taylor-made Pi." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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Keywords:    geometric series, Taylor series, pi formula
Subjects:    algebra, calculus, analysis
Level:    Advanced
Fun Fact suggested by:   Arthur Benjamin
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