
An elementary proof of Wallis’ product formula for Pi - mosca
http://fermatslibrary.com/s/an-elementary-proof-of-wallis-product-formula-for-pi
======
tzs
Another elementary proof of this is in Yaglom and Yaglom's delightful
"Challenging Mathematical Problems With Elementary Solutions (Volume 2)",
which is available as an inexpensive Dover edition [1].

It's problem #147. It asks you to deduce Wallis' formula from the identities
that problem #143 asks you to prove. Those are:

sin(pi/2m) sin(2pi/2m) sin(3pi/2m)...sin((m-1)pi/2m) = sqrt(m)/2^(m-1)

sin(pi/4m) sin(3pi/4m) sin(5pi/4m)...sin((2m-1)pi/4m) = sqrt(2)/2^m

There is also a "Challenging Mathematical Problems With Elementary Solutions
(Volume 1)", also available as an inexpensive Dover edition and, unlike Volume
2, available as en ebook [2].

Volume I's problems are divided into the following categories: introductory
problems; the representation of integers as sums and products; combinatorial
problems on the chessboard; geometric problems on combinatorial analysis;
problems on the binomial coefficients; problems on computing probabilities;
experiments with infinitely many possible outcomes; and experiments with a
continuum of possible outcomes.

Volume II's problems are divided into the following categories: points and
lines; lattices of points in the plane; topology; a property of the
reciprocals of integers; convex polygons; some properties of sequences of
integers; distributions of objects; nondecimal counting; polygons with minimal
deviation from zero (Tchebychev polynomials); four formulas for pi; the
calculation of areas of regions bounded by curves; some remarkable limits; and
the theory of primes.

These have some fun and surprising problems. For instance, problem 119 of
Volume 2: Let a=1/n be the reciprocal of a positive integer. Let A and B be
two points of the plane such that the segment AB has length 1. Prove that
every continuous curve joining A to B has a chord parallel to AB and of length
a. Show that if a is not the reciprocal of an integer, then there is a
continuous curve joining A to B which has no such chord of length a.

After the problems the books provide hints, and after the hints full solutions
are provided.

[1] [https://www.amazon.com/Challenging-Mathematical-Problems-
Ele...](https://www.amazon.com/Challenging-Mathematical-Problems-Elementary-
Solutions/dp/0486655377/ref=sr_1_1?s=books&ie=UTF8&qid=1467751087&sr=1-1)

[2] [https://www.amazon.com/Challenging-Mathematical-Problems-
Ele...](https://www.amazon.com/Challenging-Mathematical-Problems-Elementary-
Solutions/dp/0486655377/ref=pd_bxgy_14_img_2?ie=UTF8&psc=1&refRID=EDYH9A9SMH93WR3NW305)

