Talk:Existence of Radius of Convergence of Complex Power Series

The Product Rule for Sequences is invoked twice on $\limsup$s, while it hasn't been established for such quantities. In fact, in general we have $\limsup a_n b_n \le \limsup a_n \limsup b_n$ because the products may align badly (e.g. even the right hand side may be infinite and the left hand side zero, as seen from:

$a_{2n} = 2n, a_{2n+1}=0, b_{2n}=0, b_{2n+1} = 2n+1$

Equality holds as soon as one of the two actually converges). Thus the proof has to be adjusted a bit to refer to the correct theorems. --Lord_Farin (talk) 19:44, 19 January 2013 (UTC)