work on the sums of the natural number power series was largely done by
the Swiss Jacques Bernoulli (1654-1705). In English, he is sometimes
referred to as James Bernoulli, and in German, as Jakob Bernoulli.
There are probably about a dozen famous Bernoullis in mathematics or
physics, and at least two named Jacques!
Jacques had a
brother, who was also a distinguished mathematician, Jean Bernoulli
(1677-1748), also known in English as John Bernoulli and in German as
Johann. There were at least four Jean Bernoullis! According to Carl
Boyer, "they were as quick to offend as to be offended".
The Power Series As a Basis for the Bernoulli Numbers
Bernoulli studied the formulae for the sums of the powers of the
natural numbers, and noted certain facts about them. He used a longer
list than is shown below.
Some of the things that are apparent by studying the table are:
The coefficient of the first term is always 1/(m+1) (Terms begin at 0!)
The coefficient of the second term is always 1/2 (Term 1)
There seems to be a pattern to these formula. And the coefficient of the third term is m/12, for instance.
The coefficients add up to 1
Counting the first term as 0, all the odd terms, except 1, are 0.
number of terms in each sum seems to be (m+1), with any odd terms,
except 1, vanishing. So for the 7th Power, we expect 8 terms (0..7)
with the 7, 5 and 3 terms vanishing. This leaves 8-3=5. The guess, or floor(m/2)+2 seems to work for the formulae in the table (but doesn't work for zero).
While we have used a recursive formula
Bernoulli discovered a closed formula from which the sum of
a given power of natural numbers can be computed more easily, but which
requires the use of special numbers, called Bernoulli Numbers in honour
of their discoverer, Jacques Bernoulli.
Summing the Series of Natural Numbers Using Bernoulli's Formula
the formula may appear complex, it is very easy to use when computing
the powers of the natural numbers, especially, those above the third
On this page so far, we do not yet know what these Bernoulli
Numbers are. Bernoulli developed the formula by studying the series of
the various powers and realised there were certain constants that
appeared (Which he wrote as A, B, C, etc)
The series has the following pattern: that is, the formula for the sum to n terms of the m powers of the natural numbers is: [3.1]
While Bernoulli used slightly different notation, he effectively deduced the above series by observing the sums of the powers.
the B's are the Bernoulli Numbers, r is the term (0, 1, 2...). Because
the odd terms (except 1) vanish, I have not written odd number terms,
such as term 3 with B3 (which is 0). There are some observations about this formula:
The formulae produced by 3.1 end with an n term or higher power of n (no constants!).
We can immediately find some Bernoulli Numbers by comparing formula 3.1 with series above.
Except for 1, all the other odd number Bernoulli Numbers are 0.
B0=1, because all the series have 1/(m+1) as the coefficient of term0.
because in the series above, the term 1 is always 1/2. All the odd
terms in the series are negative, but only the term 1 appears (the
others vanish), so we could, for summing series make all the signs in
the formula positive, and make B1 1/2. However, we need B1 to equal -1/2 in, for instance, trigonometry.
A List of Some Bernoulli Numbers
The following is a short list of Bernoulli Numbers.
Perhaps there are two things to notice immediately. The numbers alternate between positive and negative. And B12 looks
so odd, it seems unlikely we would find a simple formula to compute
them. However, there are a number of recursive formulae, and a
relatively easy symbolic (mnemonic) method..
Examples of Using Bernoulli's Formula to Find Sums of Powers
Sum 0 Powers
If we set m=0 in the equation:
We obtain the formula with B0=1, we find:
We expected and found (0+1) terms. There is no shame in starting off simple!
Sum of Cubes
Let m=3, to find the sum of the cubes. We expect (3+1) terms, with term 3 vanishing, which leaves 3 terms.
Substituting the values for the Bernoulli Numbers: B0=1, B1=(-1/2), and B2=1/6 we get the formula for the sum of the cubes:
We note the sum of the coefficients is 1
Sum of 8th Powers
Let m=8, to find the sum of the 8th powers:
Substituting values for the Bernoulli Numbers:
And computing the values:
expected 9 terms less term 3, 5 and 7, leaving 6 terms. We expected the
sum of the coefficients to equal 1. Both these expectations were met,
giving us confidence in our algebra and arithmetic!
Generating Bernoulli Numbers
Mnemonic Trick to Easily Generate Bernoulli Numbers
The following symbolic formula can be used to generate Bernoulli Numbers:
For instance, set n=1, to find B1
Writing the powers of B in the following way, we have:
Setting n=2, for B2:
Substituting the value for B1, we find:
We can set n= (-1) to find B0, but n=0 does not work. Apart from this exception, we can use the symbolic/mnemonic formula to compute the Bernoulli Numbers.