Plane trigonometry, by S.L. Loney.

SUM OF NEGATIVE POWERS OF INTEGERS. 445
Hence, equating coefficients of 02 and 04, we have
1 1 1 I
2 32 5'2 2.
/ 1 1 1 1
4 14 34 54 + 12'...............
375. Wallis' Formula.
In the expression of Art. 369 put O=, and we have
-r1.3 3.5 5.7 (2n-3)(2n-1) (2n-.1)(2n+l)
2 22 * 42 ' 62 '''' (2n - 2)2 (2n)2
where n is infinite,
2 12. 32. 52 7......(2-).(2n+ )
and- 22. 42.6......(2)2
14 34 24..........
i.e. t2expreson of Ar (2 p +1), wheren isinfinite.
7r L * 0...... ad inf.V
It follows that when n is very great (but not necessarily infinite) then
1rl.3...... 7 (1. - 1 ) -very nearl
= n-7r, ultimately.
This is called Wallis' Formula, and gives in a simple form a very near
approach to the product of the first n even numbers divided by the first n
odd numbers when n is very great.