# Bernoulli numbers

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**Bernoulli numbers** $B_n$ are a sequence of rational numbers that satisfy the generating functions $\displaystyle \dfrac t{e^t-1} = \sum_{m=0}^\infty B_m \dfrac{t^m}{m!}.$
Bernoulli Numbers are also useful in finding the values of $\zeta(n)$ for even $n$'s.
You may try this for its application.

The values of the first few Bernoulli numbers are as follows:

$n$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |

$B_n$ | $1$ | $\pm \frac 12$ | $\frac16$ | $0$ | $-\frac1{30}$ | $0$ | $\frac1{42}$ | $0$ | $-\frac1{30}$ | $0$ | $\frac5{66}$ |

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## Properties

$B_{2k+1}=0 ~\forall k\ge 1.\ _\square$

We see that $\dfrac{t}{2}+\dfrac{t}{e^t-1}=\dfrac{t(e^t+1)}{2(e^t-1)}=\dfrac{t}{2}\coth\dfrac{t}{2}$ is an even function of $t$.

So in its power series expansion about $t = 0,$ the odd order coefficients are zero.

So $B_1+\dfrac{1}{2},B_3,B_5,B_7,\ldots$ are zero.Hence, $B_1=-\dfrac{1}{2}$ and $B_{2k+1}=0 ~\forall k\ge 1.\ _\square$

## See Also

**Cite as:**Bernoulli numbers.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/bernoulli-numbers/