This pdf gives a simple derivation of the classical formula for the sum of the positive powers of the positive integers up to a certain integer value n as a polynomial in n with rational coefficients in which the Bernoulli numbers appear in the coefficients. It derives the classical iterative formula for the Bernoulli numbers. It gives simple proofs of the facts that the polynomial for the sums of the odd powers greater than one is divisible by n2 and (n+1)2 and that for the even powers by n(n+1)(2n+1), and of the fact that the odd Bernoulli numbers of index greater than one are zero.
The ‘classical’ iterative formula for the Bernoulli numbers, given in the preceding pdf, uses all the non-zero Bernoulli numbers. This suggests that, since the higher-indexed numbers contain information about the lower-indexed numbers, it may be possible to obtain more ‘efficient’ formulas which eliminate some of the lower-indexed numbers. This pdf gives a derivation of such a formula. The method is taken from the chapter ‘Summation of Series’ of Barnard and Child (see Books page here for a brief description of this text).
A generating function is a power series whose coefficients are a sequence of interest, normally a sequence of rational numbers. This pdf gives generating functions for the Bernoulli numbers, and also for the sums of the (positive) powers of the integers. It also shows how the generating function can be used as the basis for an alternative proof that the odd Bernoulli numbers of index greater than one are zero.
In 1739 Leonhard Euler showed that the zeta function for the even positive integers, that is, the sums of the even positive integer powers of the inverses of the positive integers, can be simply expressed in terms of pi and the Bernoulli numbers. The pdf gives a straightforward derivation of this result. The result also shows incidentally that the Bernoulli numbers of the form B4k are negative and those of the form B4k+2 are positive.
This pdf gives a recursive formula for B2k as a linear combination of terms B2jB2m where m,j < k and k ≥ 2. Using the formula it demonstrates that the B4k are negative and the B4k+2> are positive.
The Euler Maclaurin Sum Formula
The Euler Maclaurin Sum Formula gives a relationship between the integral of a function over a specific range and the sum of values of the function at discrete, equidistant intervals over the range. The pdf gives a relatively straightforward derivation.
Some Properties of the Bernoulli Polynomials
The pdf derives some properties of the Bernoulli Polynomials useful in estimating the remainder term in the Euler Maclaurin Sum Formula.
Some Applications of the Euler Maclaurin Sum Formula
In spite of the ultimate divergence of the series in the formula, within just a few terms it can give remarkably accurate estimates. The pdf shows how the formula can be used to give an estimate of ζ(2) with an error of less than 10-13 and of Euler’s Constant γ with an error of less than 10-12.