I am trying to determine some necessary (and sufficient conditions) for two real valued sequences {an}, {bn}
so that the sequence [(a0/b0) + (a1/b1) + ..... + (an/bn)](bn) has a finite limit as n ---> infinity.
Any suggestions is appreciated.
[For example, if (bn) is a constant sequence, then the above implies that the series a0 + a1 + a2 + ... must converge.]