From the Fun Fact files, here is a Random Fun Fact, at the Advanced level: Taylormade Pi
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 + w^{2}  w^{3} + ...
Now subsitute x^{2} for w:
1/(1+x^{2})
= 1  x^{2} + x^{4}  x^{6} + ...
Then integrate both sides (from x=0 to x=y):
arctan y = y  y^{3}/3
+ y^{5}/5  y^{7}/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
wellknown 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. "Taylormade Pi."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
The Link for this Fun Fact:
is directly accessible here.
