Bernoulli numbers
Calculates the Bernoulli number Bn.
Bernoulli numbers Bn are defined by the Taylor expansion of the generating function x/(ex-1).
| Bn |
| \(\normalsize Bernoulli\ number\ B_n\\ (1)\ {\large\frac{x}{e^x-1}}={\large\displaystyle \sum_{\small n=0}^ {\small\infty}\frac{B_n}{n!}}x^n\\ \hspace{60px}=B_0+{\large\frac{B_1}{1!}}x+{\large\frac{B_2}{2!}}x^2+...+{\large\frac{B_n}{n!}}x^n+...\\ (2)\ B_n={\large\displaystyle \sum_{\small k=0}^ {\small n}\frac{1}{k+1}\displaystyle \sum_{\small j=0}^ {\small k}}(-1)^j\ _k C_j j^n\\ \hspace{30px} B_{2n+1}=0\hspace{15px}for\ n=1,2,...\\ (3)\ B_0=1,\ B_1=-{\large\frac{1}{2}},\ B_2={\large\frac{1}{6}},\ B_4=-{\large\frac{1}{30}},\ ...\\\) |
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