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#38
05-31-2009, 11:11 AM
 Charter Member Join Date: Jul 1999 Location: Greenbelt, Maryland Posts: 14,353
Let's start over in the explanation of Bernoulli numbers:

Look at the following sorts of sums:

0 + 1 + 2 + 3 + 4 + 5 + 6 + . . .

The sequence of sums of the nonnegative integers is the following:

0, 1, 3, 6, 10, 15, 21, 28, 36, . . .

But there's a formula for this:

Y = (X**2)/2 - X/2

(Note: X**2 means X squared. X**3 means X cubed. X**4 means X to the fourth power. And so on.)

Try it yourself. If you put 1 in for X, you get 1 for Y. If you put 2 in for X, you get 3 for Y. If you put 3 in for X, you get 6 for Y. So this means that to find out the sum of all the first X nonnegative integers, you don't have to add them up separately, you just put X into that formula and solve for Y.

Then there is the sums of the squares of the nonnegative integers:

0 + 1 + 4 + 9 + 16 + 25 + 36 + . . .

The sums are as follows:

0, 1, 5, 15, 25, 50, 86, . . .

There's a formula for this:

(X**3)/3 - (x**2)/2 + X/6

I have to define something that's slightly confusing now - the zero-th powers of numbers. The cube of a number (the third power of a number) is what you get when you multiple 1 by the number three times. The square of a number (the second power of a number) is what you get when you multiple 1 by the number two times. The first power of a number is what you get when you multiple 1 by the number one time. So the zero-th power of a number is what you get when you multiple 1 by the number zero times.

So the sequence of zero-th powers is:

1, 1, 1, 1, 1, 1, . . .

The sequence of first powers is:

0, 1, 2, 3, 4, 5, . . .

The sequence of squares is:

0, 1, 4, 9, 16, 25, . . .

The sequence of cubes is:

0, 1, 8, 27, 64, 125, . . .

O.K., so then the sum of the sequence of zero-th powers is:

1, 2, 3, 4, 5, 6, . . .

The formula for this is X + 1.

Now we're going to create another sequence. Look at the coefficient for the sum of the zero-th powers. The formula was X + 1. The coefficient for X was 1. The coefficient means the number that the variable is multiplied by in the formula.

Look at the coefficient of the X term in the formula for the sum of the first powers. The formula was Y = (X**2)/2 + X/2, so the coefficient of X was -1/2.

Look at the coefficient of the X term in the formula for the sum of the second powers. The formula was (X**3)/3 - (x**2)/2 + X/6. The coefficient for X is 1/6.

You can create an formula for any sum of the powers of integers. It turns out that the coefficient for X in the formula for cubes is 0, the coefficient for X in the formula for fourth powers is -1/30, etc.

So we have a sequence that goes like this:

1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, . . .

These numbers are the Bernoulli numbers. There are other ways of defining them, but this is the easiest way to explain it. Take this as being the definition of Bernoulli numbers.

Is this any help?
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