Finite approximations to infinite series
D.Q.M. Fay · International Journal of Mathematical Education in Science and Technology · 1989
This paper is derived from lecture notes from a course in ‘object oriented programming’. They formed part of the section comparing and contrasting Smalltalk with FP systems. The work is an extension of separate published work by Knuth [1] and Turner [2]. Three types of series are considered: (a) binomial series; (b) exponential series; and (c) recurring fractions. Series may be represented as sequences of digits in an appropriate base. The choice of the base determines the nature of the series. For example, a unary representation ofe is 1∗111111...in a variable base in which the weight of a symbol is the reciprocal of the value of that digit's position in the fraction. Such a funny base can be converted to the more familiar decimal base by iteratively multiplying by ten and using the values for the overflows into the units position. Binomial series may require the use of non‐integer bases, which can be represented using rational fractions. The same methods are used as for conversion between constant integer bases. The conversion algorithm is programmed in an object oriented language which provides the support required for sequences, function parameters and rational fractions. No use is made of floating point number representation. Termination of the algorithms is discussed fully.