The Euler Infinite Product Representation of the Factorial

I have no idea how Euler thought of it, but I did figure out how to prove it.

Citation
, XML
Authors

Abstract

To solve the problem of extending the factorial to non-integer arguments, Leonhard Euler gave a representation of the factorial as an infinite product. In this knol we give a derivation of this product.

INTRODUCTION

In the 1720’s, the best mathematicians of the day were interested in defining the factorial function for values other than nonnegative integers. Euler was the first to accomplish this; he did it by expressing the factorial as an infinite product:

n!=prod_{k=1}^{infty} frac{left( 1+frac{1}{k}  right)^n }{1+frac{n}{k} }

In doing so, he made it possible to evaluate expressions like pi!, which using the product is seen to be approximately 7.188082733.

PROOF

0!=1=prod_{k=1}^{infty} frac{1}{1} =prod_{k=1}^{infty} frac{left( 1+frac{1}{k}  right)^0 }{1+frac{0}{k} }  is clearly true, so in the following we will assume n>0.

Lemma: imageprod_{k=1}^{N} left( left( 1+frac{1}{k}  right) left( 1-frac{1}{n+k}  right)  right) =frac{N+1}{N+n} n

Proofprod_{k=1}^{N}left(left(1+frac{1}{k}right) left( 1-frac{1}{n+k}  right)right)=left( prod_{k=1}^{N}left(1+frac{1}{k}right)right)left(prod_{k=1}^{N}left(1-frac{1}{n+k}right)right)

=left( prod_{k=1}^{N} frac{k+1}{k} right)   left( prod_{k=1}^{N} left( 1-frac{1}{n+k}  right) right)  = frac{N+1}{1}   prod_{k=1}^{N} left( 1-frac{1}{n+k}  right)

= frac{N+1}{1}   prod_{k=1}^{N}frac{n+k-1}{n+k}  = frac{N+1}{1}   frac{n}{n+N} =frac{N+1}{N+n} nqed

image

Lemma: imageprod_{k=1}^{N} frac{left( 1+frac{1}{k}  right)^n }{1+frac{n}{k} } =n!prod_{j=1}^{n} frac{N+1}{N+j}

Proofprod_{k=1}^{N} frac{left( 1+frac{1}{k}  right)^n }{1+frac{n}{k} } =prod_{k=1}^{N}left( left( 1+frac{1}{k}  right)^n frac{k }{k+n}right)

=prod_{k=1}^{N}left(left(1+frac{1}{k}right)^nprod_{j=1}^{n}frac{j+k-1}{j+k}right)=prod_{k=1}^{N}left(left( 1+frac{1}{k}  right)^n  prod_{j=1}^{n} left( 1-frac{1}{j+k}  right) right)

=prod_{k=1}^{N}prod_{j=1}^nleft(left(1+frac{1}{k}right)left(1-frac{1}{j+k}right)right)= prod_{j=1}^nprod_{k=1}^{N}left(left(1+frac{1}{k}right)left( 1-frac{1}{j+k}  right)  right)

= prod_{j=1}^n left( frac{N+1}{N+j} j right) = n!prod_{j=1}^n frac{N+1}{N+j}qed

image

Theorem: imageprod_{k=1}^{infty} frac{left( 1+frac{1}{k}  right)^n }{1+frac{n}{k} } =n!

Proofprod_{k=1}^{infty} frac{left( 1+frac{1}{k}  right)^n }{1+frac{n}{k} } =lim_{N rightarrow infty}prod_{k=1}^{N} frac{left( 1+frac{1}{k}  right)^n }{1+frac{n}{k} }

=lim_{N rightarrow infty} left( n! prod_{j=1}^{n}  frac{N+1}{N+j}   right) =n! lim_{N rightarrow infty}  prod_{j=1}^{n}  frac{N+1}{N+j}  =n!qed

Notice that any positive value of n can be substituted in the infinite product, not just nonnegative integers. This representation led immediately to the very important gamma function.