Ken Ward's Mathematics Pages

Series Sums of the Powers of the Natural Numbers and Bernoulli Numbers

Series Contents

Page Contents

1. A Note on the Discoverer of the Bernoulli Numbers
2. The Power Series As a Basis for the Bernoulli Numbers
3. Summing the Series of Natural Numbers Using Bernoulli's Formula
4. A List of Some Bernoulli Numbers
5. Examples of Using Bernoulli's Formula to Find Sums of Powers
6. Generating Bernoulli Numbers
1. Mnemonic Trick to Easily Generate Bernoulli Numbers

A Note on the Discoverer of the Bernoulli Numbers

The 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

Jacques 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.

Power	m+1	m	m-1	m-2	m-3	m-4	m-5	Formula
1	1/2	1/2
2	1/3	1/2	1/6
3	1/4	1/2	1/4
4	1/5	1/2	1/3		-1/30
5	1/6	1/2	5/12		-1/12
6	1/7	1/2	1/2		-1/6		1/42
7	1/2	7/12			-7/24		1/12
8	1/2	2/3	-7/15		2/9		-1/30

Some of the things that are apparent by studying the table are:

1. The coefficient of the first term is always 1/(m+1) (Terms begin at 0!)
2. The coefficient of the second term is always 1/2 (Term 1)
3. There seems to be a pattern to these formula. And the coefficient of the third term is m/12, for instance.
4. The coefficients add up to 1
5. Counting the first term as 0, all the odd terms, except 1, are 0.
6. The 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

Although the formula may appear complex, it is very easy to use when computing the powers of the natural numbers, especially, those above the third power.

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.

Where 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.

1. Except for 1, all the other odd number Bernoulli Numbers are 0.
2. B₀=1, because all the series have 1/(m+1) as the coefficient of term0.
3. B₁=-1/2, 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.
4. B₂=1/6
5. B5=-1/30
6. B₇=1/42

A List of Some Bernoulli Numbers

The following is a short list of Bernoulli Numbers.

B0	B1	B2	B4	B6	B8	B10	B12
1	-1/2	1/6	-1/30	1/42	-1/30	5/66	-691/2730

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:

[3.1, repeated]

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:

We 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:


Or B₁=-1/2

Setting n=2, for B₂:


Substituting the value for B₁, we find:

Giving B₂=1/6

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.















Ken Ward's Mathematics Pages

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Faster Arithmetic - by Ken Ward