Bernoulli Numbers

I recently learnt about bernoulli numbers in wikipedia . But i cannot understand what is the basic definition for these type of numbers . Can anyone give a short description about these numbers and the definition of them ?

#Bernoullinumbers

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Jakob Bernoulli discovered these numbers while investigating finite sums of powers, which can always be represented by a finite polynomial with rational coefficients. To cut to the chase, here's how it comes out, where ${ B }_{ k }$ is the $k$ th Bernoulli number:

$\displaystyle \sum _{ k=1 }^{ n }{ { k }^{ m } } =\dfrac { 1 }{ m+1 } \sum _{ k=0 }^{ m }{ \left( \left( \begin{matrix} m+1 \\ k \end{matrix} \right) { B }_{ k } { n }^{ m+1-k } \right) }$

Bernoulli numbers pop up in a great many other places in mathematics, and many other generating functions have been devised for these numbers. But the formula given above was the original definition, as given by Jakob Bernoulli.

Michael Mendrin - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$