# Bernoulli number Calculator

## Calculates the Bernoullis numbers Bn .

 Bn is a coefficient of the nth term of Taylor expansion of the generating function x/(ex-1).
degree n
 n=0,1,2,...

Bn
 $\normal Bernoulli\ number\ B_n\\[10](1)\ {\large\frac{x}{e^x-1}}={\large\sum_{\small n=0}^ {\small\infty}\frac{B_n}{n!}}x^n\\\hspace{60}=B_0+{\large\frac{B_1}{1!}}x+{\large\frac{B_2}{2!}}x^2+...+{\large\frac{B_n}{n!}}x^n+...\\[10](2)\ B_n={\large\sum_{\small k=0}^ {\small n}\frac{1}{k+1}\sum_{\small j=0}^ {\small k}}(-1)^j\ _k C_j j^n\\[10]\hspace{30} B_{2n+1}=0\hspace{15}for\ n=1,2,...\\[10](3)\ B_0=1,\ B_1=-{\large\frac{1}{2}},\ B_2={\large\frac{1}{6}},\ B_4=-{\large\frac{1}{30}},\ ...\\$

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