Bernoulli number

Bernoulli numbers are almost the sum of k power.
Bn is the Bernoulli number. This is recurrence relation, so you change the direction back and forth.

You can also write this.

This is Maclaurin's expansion.
Then you can see this.

f(x)*(1/f(x))=1

This is convergence.

∴

from (2)

This is Binomial Coefficient.

∴

EX.
B0=1
B1=-1/2*(2C0)*B0=(-1/2)*1*1=-1/2
B2=-1/3*(3C0*B0+3C1B1)=(-1/3)*(-1/2)=1/6
B3=-1/4*(4C0*B0+4C1*B1+4C2*B2)=-1/4*(1-2+1)=0
B4=-1/5*(5C0*B0+5C1*B1+5C2*B2+5C3*B3)=(-1/5)*(1/6)=-1/30

All Bernoulli number is rational.
Bn is related to Riemann zeta function.

n→∞