Bill Dubuque has answered several questions by indicating that some form of "telescoping" is taking place. See this post and the links provided by Bill for more information.
I have never heard of "telescoping" until I read a few answers on here by Bill which refer to this notion. It seems fairly straightforward, basically you expand some expression using basic arithmetic, there is a minor miracle and lots of terms cancel out in a systematic way, and we are then able to solve the problem.
I suppose "telescoping" in this sense was something I always took for granted, and considered a low level "trick" to keep in my back pocket. However, considering the importance Bill seems to attach to this notion of telescoping, and considering that I have a great deal of respect for Bill based on the post's by him I have read, I was wondering if I'm not missing something about telescoping. There is no wiki article on the subject, and a Google search directs me to Bill's answers on SE!
Therefore I would like to ask:
1) What unintuitive results can I achieve with telescoping?
2) Is there a good reference which only discusses telescoping and applications, or is this concept too narrow for anyone to write a paper/book like this?
3) More trivially, am I missing something about what telescoping actually means? If not, then why is this called telescoping, because I don't see what this has to do with a telescope?