A New Approach to Evaluation of Infinite Processes
Abstract
The simplest forms of discrete infinite processes, such as infinite series, products, continued fractions and their generalizations are considered. It is shown that by associating such processes with "equivalent" linear difference equations with boundary conditions at infinity a means of classifying them in a unified way is provided, as well as a means of evaluating asymptotic approximations to remainder sequences. If the approximate remainder sequences are introduced at the definitional level, so that the "value" of the infinite process is defined as a limit of successive stages of the finite process with an approximate remainder term included at each stage, two benefits result. First, where the process converges by the Cauchy definition (zero remainder terms), convergence is speeded, so that numerical computations of 'value' are aided. Secondly, where the process is Cauchy-divergent, it may nevertheless be 'summed' to a useful value. A broad class of processes, termed "asymptotically tractable," is identified for which these benefits are obtained.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1971
- Accession Number
- AD0723555
Entities
People
- Thomas E. Jr Phipps
Organizations
- Naval Ordnance Laboratory