Bernoulli and Euler's Numbers

Definition. The Bernoulli numbers are defined by

\begin{displaymath}\begin{array}{lclrl}
\displaystyle \frac{x}{e^x - x} &=& 1 - ...
...B_3x^6}{6!}+ \cdots &\mbox{$\vert x\vert < \pi$}\\
\end{array}\end{displaymath}




Definition. The Euler numbers are defined by

\begin{displaymath}\begin{array}{lccclrl}
\mbox{sech}(x) &=& \displaystyle \frac...
...E_3x^6}{6!}+ \cdots &\mbox{$\vert x\vert < \pi$}\\
\end{array}\end{displaymath}




Some Important Formulas.

1.
$B_n = \displaystyle \frac{(2n)!}{2^{2n-1}\pi^{2n}} \left(1 + \displaystyle \frac{1}{2^{2n}} + \displaystyle \frac{1}{3^{2n}} + \cdots \right)$

2.
$B_n = \displaystyle \frac{2(2n)!}{(2^{2n-1}-1)\pi^{2n}} \left(1 - \displaystyle \frac{1}{2^{2n}} + \displaystyle \frac{1}{3^{2n}} - \cdots \right)$

3.
$E_n = \displaystyle \frac{2^{2n+2}(2n)!}{\pi^{2n+1}} \left(1 - \displaystyle \frac{1}{3^{2n+1}} + \displaystyle \frac{1}{5^{2n+1}} - \cdots \right)$

4.
For large n, we have

\begin{displaymath}B_n \sim 4 n^{2n} (\pi e)^{-2n} \sqrt{n\pi} = 4 \left(\displaystyle \frac{n}{\pi e}\right)^{2n} \sqrt{n \pi}\end{displaymath}

Below you may find some values of the Bernoulli numbers:



and Euler numbers



[Tables]

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