I have finally got an answer from http://www.allexperts.com/el/Number-Theory/ to my OP question from an expert ( Scott A Wilson ) . The answer he gave was:
"...That is an infinite sum, so there is no limit.
The individual terms go to 0, but the sum does not.
Approximating this with an integral gives the integral of 1/x for x going from 1 to infinity.
Since this is ln(x), the ln() of infinity is infinity, so there is no limit.
The value of n can never be infinity, but it only tends towards infinity.
No matter what value is given to n, n+1 is greater.
That is why it is said to be the limit as n tends to infinity.
At first, I thought that sounded contradictory because the "no limit" and "infinite" parts in the
"That is an infinite sum, so there is no limit."
made it sound to me that n CAN be infinite (because how can it be an "infinite sum" if n is finite? ) , but then he said;
"The value of n can never be infinity,"
which contradicted that.
But then the thought occurred to me that "no limit" doesn't necessarily imply "infinity"! Because "no limit" in this context means, as I think he implied by, "No matter what value is given to n, n+1 is greater.", you cannot define any specific finite limit because there is none but that doesn't logically imply n can be let alone is infinite!
-am I thinking about that in the right way?
And does anyone here disagree with him that n cannot literally be infinite?