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 ?

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.

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TopNewestJakob 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.

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