THE CALCULUS ANSWER (FOR THE BOLD)
One studies infinite sums in a calculus class. You learn there, for example, that is true as a statement of arithmetic for small values of , (specifically, for all values of strictly between and .) The formula is valid for , as we saw, and not for .
For those who are game, here is how the calculus argument goes.
Regular polynomial division shows that and and , and so on. (Try these!)
In general, we see that
Now as we let get bigger and bigger it looks like we’re getting the infinite geometric sum.
So the question is: What does go to as gets bigger and bigger? If there is an answer to this question, then the answer will be the value of .
So does have a limit value? Well, this depends on whether or not has a limit value as grows. So for which values of do the powers of it approach a value?
We know the that powers of , for example, and of and of , all approach zero for bigger and bigger powers. In fact, gets closer and closer to zero as grows for any value between and .
So for , we have
The geometric series formula can be believed, as a statement of arithmetic, for , at least.