Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

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>.

Keywords:    geometric series, Taylor series, pi formula
Subjects:    algebra, calculus, analysis
Level:    Advanced
Fun Fact suggested by:   Arthur Benjamin
Suggestions? Use this form.
3.77
 
current
rating
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
Get the Math Fun Facts
iPhone App!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!