Asymptotic Formula for Bernoulli Numbers

Theorem
The Bernoulli numbers with even index can be approximated by the asymptotic formula:


 * $B_{2 n} \sim \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\dfrac n {\pi e} }^{2 n}$

where:
 * $B_n$ denotes the $n$th Bernoulli number
 * $\sim$ denotes asymptotically equal.

Also rendered as
This result can also be seen expressed as:


 * $B_{2 n} \sim \paren {-1}^{n + 1} 4 n^{2 n} \paren {\pi e}^{-2 n} \sqrt {\pi n}$