13 - Ramifications of the Binomial Theorem

Abstract

There are many applications and extensions of the binomial theorem. This chapter discusses several ramifications of the binomial theorem or some applications and extensions of the binomial theorem. It also described the other ways in which binomial-type expressions occur. Kenneth Berman applies the binomial theorem to arithmetic power series. The chapter further discusses a connection between the binomial theorem and probability, and the binomial theorem in a discussion of the distribution of objects into boxes. In arithmetic power series, one may develop a recursion formula for summing an arithmetic power series, a series of the form Snk = ak + (a + d)k + (a + 2d)K + + (a + nd)k in which the terms on the right which are raised to the kth power are just the terms of the arithmetic series: Sn = a + (a + d) + (a + 2d) + · + (a + nd). This development illustrates at least four important techniques: (i) the symbolic approach, (ii) the use of the binomial theorem, (iii) the use of recursion relations, and (iv) the summation method.