You're objecting to his 2009 paper, which is about a sum over infinite time. This 2007 paper is timeless.

My response to his 2009 paper is here. At present I think that it a) assumes what it is trying to prove, and b) assumes there is no time discounting, and so could be seen as an argument for time discounting. But I haven't looked at it closely enough to be confident of (a), given how accurate Peter typically is.