I first encountered the Bernoulli numbers in a combinatorics class, and they came up later in asymptotic analysis in the Euler-Maclaurin Formula.  There are also Bernoulli polynomials as well.

To compute the Bernoulli number, usually denoted by in mathematics, in MathStudio, use Bernoulli(n).  For example, the first Bernoulli number is given in MathStudio by .  There is some discussion as to whether this first value should be taken to be positive or negative, and on the wikipedia page both versions are explored fully.  Personally I’ve seen the negative value used in my applications.

Bernoulli numbers have a rich set of applications, my favorite being Bernoulli’s formula, which is

This formula basically shows how to take the sum of powers of the first integers and get an exact expression.  By looking at the dominant term, we see that

,

which basically means that the dominant term is the first term, which agrees with our intuition that integrals are continuous versions of summations.

More generally, the Euler-Maclaurin formula governs the relationship between a sum and an integral for smooth enough functions ,

,

where is a remainder term that depends on the value of , the order of the expansion.

Also of interest are Bernoulli polynomials, which are related to Bernoulli numbers.  In MathStudio, to obtain the Bernoulli polynomial in the variable , just add an argument from before, Bernoulli(n,x).

Exercise 1. Give an exact expression for the sums , , , .

Exercise 2.  Use Exercise 1 to show that .