Abstract
In an article* in this MONTHLY the of slowly convergent continued fractions was suggested by the author as a means of obtaining forms that would converge more rapidly and thus be more useful in computation. In the present article the general considerations underlying such are developed and the procedure is illustrated with some concrete examples. The notion of a modified continued fraction is not new; Sylvestert in 1869 called attention to a first modification of Euler's continued fraction for wT/2, and Glaishert in 1874 compared this result with a similarly modified Wallis product. But its usefulness seems not to have been appreciated. 1. Definition. Given the convergent infinite continued fraction bo+aj/bl +a2/b2 + * 1, by Kl,=A,/B, we denote the convergent of order v, and by Lv= Cl/Dv, the same finite continued fraction with the last partial quotient, avl/b, replaced by cl/dv. The problem is to determine cl/dv so that (a) limvOO Lv=lim,OO Kv, (b) L, will converge more rapidly than Kv.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
