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WHAT ARE THE BERNOULLI NUMBERS?

WHAT ARE THE BERNOULLI NUMBERS? C. D. BUENGER. Abstract. For the "What is?" seminar today we will be investigating the BERNOULLI numbers. This surprising sequence of numbers has many applications including summing powers of integers, evaluating the zeta function, finding asymptotics of Stirling's formula, and estimating the harmonic series. We investigate properties of these numbers and introduce BERNOULLI polynomials, a closely related topic. Then, we establish the Euler summation formula and use the formula to provide the aforementioned applications. Furthermore, a study of the Fourier coefficients, yields surprising results on the zeta function.

2 is even, and x is odd. Thus x ex1 B 1x is an even function. Thus the power series expansion of x ex1 B 1x has no nontrivial odd terms. The Bernoulli numbers grow quite quickly. Indeed, we will show in section 5 that B k ...

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