In mathematics, Wallis' product for π, written down in 1655 by John Wallis, states that
Contents |
Wallis derived his product as it is done in calculus books today, by comparing for even and odd values of n, and noting that for large n, increasing n by 1 makes little change. Since infinitesimal calculus as we know it did not yet exist then, and mathematical analysis sufficient to discuss the convergence issue was inadequate, this was a harder piece of research than it sounds with hindsight, and more tentative. Wallis's product is, in retrospect, an easy corollary of the later Euler formula for the sine function.
Let x = π/2:
Let:
Repeating the process,
Repeating the process,
By the squeeze theorem,
Stirling's approximation for n! asserts that
as n → ∞. Consider now the finite approximations to the Wallis product, obtained by taking the first k terms in the product:
pk can be written as
Substituting Stirling's approximation in this expression (both for k! and (2k)!) one can deduce (after a short calculation) that pk converges to π/2 as k → ∞.
The Riemann zeta function and the Dirichlet eta function can be defined:
Applying an Euler transform to the latter series, the following is obtained: