You need to edit your comment. Not enough back ticks, and your 2nd formula is missing.
The problem with infinity and asymptotic is that they rely on the idea that you can always add one to a number.
In real life you can't. Take any set of real objects, such as all the atoms in the universe. And, although that is a very big number, it is not infinity. And if we are talking about atoms, there is no meaning to one more, when you have all. And math being a tool for expressing physics, it shouldn't give us meaningless results.
But, this would imply that the series above go to 15/16 or to 6. Two very real numbers. To go outside that bounds just ends up in meaninglessness.
Further, there may be a number that is designed as the maximum number in this universe.
Kind of like the plank constant. There is no zero, there is no approaching zero, there is the plank constant or nothing.
If there is such a number, then we need to calibrate our maths to that number.