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:

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$B_n$	$1$	$\pm \frac 12$	$\frac16$	$0$	$-\frac1{30}$	$0$	$\frac1{42}$	$0$	$-\frac1{30}$	$0$	$\frac5{66}$

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

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Contents

Properties

See Also