Notes on Explorations 413For a general introduction on the nature of the real number field, which
includes a chapter on continued fractions, see Ivan Niven, Irrational num-
bers (Carus monographs ll), (Math. Assoc. of Amer., 1956, 1967).
A definitive work is H.S. Wall, Analytic Theory of Continued Fractions,
(D. van Nostrand, 1948).
E.49. Interest in the iterations of the function f(z) = az( 1 -z) has greatly
increased during the last decade with the study of chaotic behaviour and
the advent of high speed computers capable of dealing with complex prob-
lems. For a gentle introduction, see A.K. Dewdney, Probing the strange at-
tractions of chaos (Computer Recreations), Scientific American 257 (#l)
(1987) 108-111.
A recent book which explores the visual beauty of this area of mathe-
matics is H.-O. Peitgen & P.H. Richter, The Beauty of Fractals: Images of
Complex Dynamical Systems, (Springer-Verlag, 1986).
A captivating layman’s introduction to this new branch of mathematics
is the book James Gleick, Chaos: Making a New Science, (Viking Penguin,
1987). This book also recounts the story of the Mandelbrot set, introduced
in Exploration E.67.
E.54. The formula for the sum of the first n kth powers for small values
of k are given by the following formulae:k sum of first n kth powers1 n(n + 1)/2 = n2/n + n/2
2 n(n + 1)(2n + 1)/6 = n3/3 + n2/2 + n/6
3 n2(n + 1)2/4 = n4/4 + n3/2 + n2/4
4 n5/5 + n4/2 + n3/3 - n/30
5 n6/6 + n5/2 + 5n4/12 - n2/12
6 n7/7 + n6/2 + n5/2 - n3/6 + n/42
7 nd/8 + n7/2 + 7n6/12 - 7n4/24 + n2/12The coefficients involve a special sequence of numbers called the Bernoulli
numbers. For some exercises on this topic, consult M. Spivak, Calculus (2nd
ed., Publish or Perish, Washington, 1980), Exercise 7, p. 29-30; Exercises
16, 17, p. 538-541.
These sum formulae were derived by Jakob Bernoulli in his book, Ars
conjectandi, published in 1713. For an English translation of the relevant
excerpt, consult pages 316-320 of D.J. Struik (ed.), A Source Book in Math-
ematics, 1200-1800, (Harvard, 1969); or pages 85-90 of D.E. Smith, A
Source Book in Mathematics, Volume One, (Dover, 1959).
An elementary derivation can be found in John G. Christiano, On the
sum of powers of natural numbers, Amer. Math. Monthly 68 (1961), 149-
151; and in Dumitru Acu, Some algorithms for the sums of integer powers,
Math. Mug. 61 (1988), 189-191 are obtained other identities involving these