What does the series: sum from j=0 to infinity of [λt/(1+λt)]^(2*j) converge to?

This is a geometric series. The first term is a = 1 and the common ratio is r = [λt/(1+λt)]^2. Recall that the sum of a geometric series is S = a/(1-r), provided that |r| < 1. If |r| ≥ 1 then the series diverges.