Meaning of Bernoulli number | Babel Free

Definitions

Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions.

Examples

“1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418, The assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely t/(eᵗ-1)=∑ₙ₌₀ ᪲B_n(tⁿ)/(n!)”

“In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.”

“This done, he extends occurrence of Bernoulli numbers in the expansion of #92;textstyle#92;frac 1 2#92;cot(#92;frac 1 2x) to the more general form #92;textstyle#92;frac#92;pin#92;cot(#92;frac#123;m#92;pi#125;n) and uses that to relate Bernoulli numbers to the values of #92;zeta(2n). To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series.”

CEFR level