According to my calculus book two parts to testing an alternating series for convergence. Let s = Ʃ(-1)n bn. The first is that bn + 1 < bn. The second is that the limn[itex]\rightarrow[/itex]∞ bn = 0. However, isn't the first condition unnecessary since bn must be decreasing if the limit is zero. I can't think of any example where the limit will be zero and the function is increasing (assuming bn is not negative which would not really make sense since the series is alternating).