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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 Bk in the following equation:bernoulliNumbersDefinition.gif [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:
e^x-1.gif [1.2]

And, expanding the RHS:
bernoulliNumbers2.gif [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 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.
bernoulliNumbers1.gif [1.4]

Expanding both sides:
bernoulliNumbers3.gif [1.5]

Now we can equate coefficients:
equateCoefficients1.gif [1.5.1]x
equateCoefficients2.gif [1.5.2]x2
equateCoefficients3.gif [1.5.3]x3
equateCoefficients4.gif [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 Bn, we have:
equateCoefficients6.gif [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+1Ck) in order to get a formula for generating the Bernoulli Numbers:
bernoulliFormulaGenNumbers.gif [1.7]

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

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