Bernoulli Numbers are defined as those Bk in the following equation: [1.1] We
need to expand the left-hand side and equate the coefficients of the
power series with corresponding coefficients of the right-hand side.
first sight, it seems difficult to expand the left-hand side into a
power series using Maclaurin's expansion. In fact there are many ways
of expanding the left-hand side and we will mention two of them.
Actually, we need to derive a formula which makes the calculation
easier, or we can use the symbolic/mnemonic method.
In either case we need: [1.2]
And, expanding the RHS: [1.3]
first method we could use is to divide x in 1.1 by 1.2, using
normal polynomial division. What we need to remember is that we need to
have sufficient terms of ex-1 to get our required coefficients.
The second method being mentioned is simply to multiply the RHS of 1.1 by (ex-1), and equate the coefficients of the x's. [1.4]
Expanding both sides: [1.5]
Now we can equate coefficients:
we kept all our factorials, a patterns becomes evident. Except
for 1.5.1 we have the formula, which can be used to generate the
Bernoulli Numbers. In the formula below, n is the highest Boolean
number in the series (power of x is n+1). So for a series ending in Bn, we have: [1.6]
convenience, we can multiply throughout by (n+1)! (because we see how the
bottom factorials seem to be related to a combination, n+1Ck) in order to get a formula for generating the Bernoulli Numbers: [1.7]