On the convergence of limit periodic continued fractions K(a(n)/1) where a(n)->-1/4. Part IV

Continued fractions K (a(n)/b(n)), where a(n), b(n) is an element of C and an/b(n)b(n-1) --> -1/4, may converge or diverge depending on how a(n)/b(n)b (n-1) approaches its limit. Due to equivalence transformations it suffices to study the special case where all b(n) = 1. We shall prove that K(a(n)/1) converges if a(n) --> - 1/4 and there exists a set V subset of or equal to C boolean OR {infinity} 4 with certain properties such that a(n)/(1 + V) s ubset of or equal to V for all n. We shall also summarize some other useful consequences of such value sets V.