Uh oh. I just realized that my choice of phrasing in my initial post in this thread has been less than optimal for clear communication. Multiple times in that post I said the phrase "finite telescoping sum" or "telescoping sum". This is misleading for what I was trying to communicate. What I should have said was "finite geometric sum/progression" or "geometric sum" in those places. The standard concept of a telescoping sum is different than the idea of a summable geometric sum of terms whose sum can be written as a quotient of differences. Typically a telescoping sum is one where subsequent terms act to cancel previous terms. I did not mean that latter standard usage in my post. I hope I didn't excessively confuse people with that above unfortunate choice of phraseology.