KEYWORDS: Bernoulli numbers, asymptotic inclusion, asymptotic approximation
OEIS A000367, OEIS A027641.

An approximation of the Bernoulli numbers.

There is a standard asymptotic formula for the Bernoulli numbers. For example at Mathworld or on the web pages of X. Gourdon and P. Sebah one can find the formula

bernaasy

I found two inclusions for the Bernoulli numbers which appear to be new. I will report these inclusions on this page. The bounds given comprise a simple but amazingly efficient approximation to the Bernoulli numbers which I will call the 'cute approximation'.

A sharp inclusion of the Bernoulli numbers.

Let Bn denote the Bernoulli numbers. If n is even and n >= 38 then

berinklusion

For example the inclusion predicts

0.5318704469415522033..*10^1770 < |B(1000)| < 0.5318704469415522039..*10^1770.

And indeed |B(1000)| = 0.5318704469415522036..*10^1770.

Note that the factorial function is not referenced in these formulas.

A cute approximation of the Bernoulli numbers.

The lower bound of the above inclusion can also be used as a convenient approximation for the Bernoulli numbers. It then takes the form (assuming n even)

bernapprox

 Equivalently this formula can be stated as

bernapprox

For example the standard approximation gives B(1000) ~ 0.53182..*10^1770, which amounts to 4 valid decimal numbers, whereas the last approximation gives for B(1000) an approximation with almost 18.2 valid decimal digits.

A sharper inclusion of the Bernoulli numbers.

Even more powerful and convenient is the following asymptotic approximation of the Bernoulli numbers.

bernapp2

bernapp

The first five terms of an stem from Stirling's formula (HMF 6.1.37).

Let n be an even integer. Choosing X = 0 in the above formula gives a low bound for Bn for n >= 2. Choosing X = 1 in the above formula gives a high bound for Bn for n >= 44. Choosing X = 1/2 will give an approximation for Bn which is superior to the one given in the last paragraph.

For example for n = 1000 and X = 1/2 gives

  |B(1000)| ~ 0.53187044694155220364823..*10^1770, compared to
  |B(1000)| = 0.53187044694155220364829..*10^1770.

This amounts to an approximation with almost 22 valid decimal digits.

Maple code snippet.

a:=(s,X)->(1+1/12/s+1/288/(s^2)-139/51840/(s^3)-571/2488320/(s^4)+(24+X)/31250/(s^5));
absBernoulli := (s,X) -> 4*Pi*exp(1/2)*(s/(2*Pi*exp(1)))^(s+1/2)*a(s,X);

# Test
Digits:=32:LOW:=0:MID:=1/2:HIG:=1:s:=600;
abs(bernoulli(s)):   B:=evalf(%);
absBernoulli(s,LOW): L:=evalf(%);
absBernoulli(s,MID): M:=evalf(%);
absBernoulli(s,HIG): H:=evalf(%);
is(L<B),is(B<H);

Note: The first inclusion was given on Jan. 18, 2007 by Peter Luschny. Here his announcement in the newsgroup de.sci.mathematik. The sharper inclusion was given by Peter Luschny on Jan. 20, 2007, on this web page.

previous Asymptotic inclusions and approximations for the Euler numbers.
next To the homepage of factorial functions.