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

An approximation of the Euler numbers.

There is a standard asymptotic formula for the Euler numbers. For example at Mathworld one can find the formula

I found two inclusions for the Euler 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 Euler numbers which I will call the 'cute approximation'.

A sharp inclusion of the Euler numbers.

Let E_n denote the Euler numbers. If n is even and n >= 20 then

For example the inclusion predicts

0.3887561841253070612..*10^2372 < |E(1000)| < 0.3887561841253070617..*10^2372.

And indeed |E(1000)| = 0.3887561841253070615..*10^2372.

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

A cute approximation of the Euler numbers.

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

Equivalently this formula can be stated as

For example the standard approximation gives E(1000) ~ 0.38872..*10^2372, which amounts to 4 valid decimal numbers, whereas the last approximation gives for E(1000) an approximation with almost 18.2 valid decimal digits.

A sharper inclusion of the Euler numbers.

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

The first five terms of a_n 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 E_n for n >= 2. Choosing X = 1 in the above formula gives a high bound for E_n for n >= 24. Choosing X = 1/2 will give an approximation for E_n

which is superior to the one given in the last paragraph.

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

  |E(1000)| ~ 0.38875618412530706152569...*10^2372, compared to
  |E(1000)| = 0.38875618412530706152573...*10^2372.

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

Maple code snippet.

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

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

Note: The 'cute approximation' was given on Jan. 21, 2007 by Peter Luschny. Here his announcement in the newsgroup de.sci.mathematik. The two inclusions were given by Peter Luschny on Jan. 22, 2007, on this web page.

Asymptotic inclusions and approximations for the Bernoulli numbers.
To the homepage of factorial functions.