AN ELEMENT AR Y PR OOF OF W ALLIS’ PR ODUCT F ORMULA

F OR PI

JOHAN W

¨

ASTLUND

Abstract. W e giv e an elementary proof of the W allis pro duct formula for pi.

The proof do es not require any in tegr ation or trigon ome tric functio ns.

1. The W allis product f ormula

In 1655, John W allis wrote down the celebrated form ula

(1)

2

1

·

2

3

·

4

3

·

4

5

··· =

⇡

2

.

Most textb ook pro ofs of (1) rely on ev aluation of some deﬁnite in tegral like

Z

⇡ / 2

0

(sin x )

n

dx

b y repeated partial in tegration. The topic is usually reserv ed for more adv anced

calculus courses. The purpose of this note is to sho w that (1) can be derived

using only the mathematics taugh t in elementary school, that is, basic algebra, the

Pythagorean theorem, and the form ula ⇡ · r

2

for the area of a circle of radius r .

Viggo Brun gives an accoun t of W allis’ metho d in [1] (in Norwegian). Y aglom

and Y aglom [2] give a b eautiful pro of of (1) whic h av oids integration but uses some

quite sophisticated trigonometric iden tities.

2. A number sequence

W e denote the W allis product by

(2) W =

2

1

·

2

3

·

4

3

·

4

5

··· .

The partial products in v olving an ev en num b er of factors form an increasing se-

quence, while those in volving an o dd n um b er of factors form a decreasing sequence.

W e let s

0

= 0, s

1

= 1, and in general,

s

n

=

3

2

·

5

4

···

2 n  1

2 n  2

.

The partial pro ducts of (2) with an o dd num b er of factors can b e written as

2 n

s

2

n

=

2

2

· 4

2

··· (2 n )

1 · 3

2

··· (2 n  1)

2

>W ,

while the partial pro ducts with an even n um b er of factors are of the form

2 n  1

s

2

n

=

2

2

· 4

2

··· (2 n  2)

2

1 · 3

2

··· (2 n  3)

2

· (2 n  1)

<W .

It follo ws that

(3)

2 n  1

W

<s

2

n

<

2 n

W

.

Date : F ebruary 21, 2005.

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