Ken Ward's Mathematics Pages

Series Bernoulli Numbers

Series Contents
See also:
Mnemonic Trick to Easily Generate Bernoulli Numbers

Page Contents

1. Definition of the Bernoulli Numbers

Definition of the Bernoulli Numbers

Bernoulli Numbers are defined as those B_k 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.

At 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]

The 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 e^x-1 to get our required coefficients.

The second method being mentioned is simply to multiply the RHS of 1.1 by (e^x-1), and equate the coefficients of the x's.
[1.4]

Expanding both sides:
[1.5]

Now we can equate coefficients:

	Equating
[1.5.1]	x
[1.5.2]	x2
[1.5.3]	x3
[1.5.4]	x4

Because 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 B_n, we have:
[1.6]

For convenience, we can multiply throughout by (n+1)! (because we see how the bottom factorials seem to be related to a combination, ⁿ⁺¹C_k) in order to get a formula for generating the Bernoulli Numbers:
[1.7]

When n=0, the sum is not zero, but 1, as in 5.31.














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Faster Arithmetic - by Ken Ward