I am extremely confused right now, concerning the limits and I guess the limit rules... I'm either reading it wrong or its just not explaining it well enough for me to understand. Basically, I am lost on this entire chapter of limits. This is kinda what I think is going on. There are limits to finite series and a finite series is a series of numbers that continue up until a stopping point or number, like 1,2,3,4...10. Then there are infinite series, which never end, but converge to a number, which I guess would be infinite, so in my mind, these partial sums, which are always getting smaller(?), because they are fractions(?), will never reach this point, and become an infinitesimal, which is a number so small that Thompson says it can be 'thrown out'. However, it says in this book that these numbers can go beyond this 'limit', somehow, in an infinite series. My first question is how is this possible? My understanding of this whole limit of infinite series is (and is probably wrong, because I'm totally confused), is that lets say I have a square. Corner A of this square to corner B of this square is 10 yards. This 10 yards is the limit, of an infinite number of squares in a row, perhaps. So first I go, 5 yards, then 2.5 yards, etc. I will never reach this particular SQUARE'S limit..but it doesn't or isn't relevant to the infinite number of squares because we are solving whatever it is we're trying to solve for this one particular 'limit' square. So is that at least partly right? Lastly, the book gave me some examples of what the Gardner calls, "the integral limit of any repeating decimal." So my last question is what is the point of solving this integral limit of any decimal?