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The set of all integers is infinite we can set n as infinite. If we set n infinite, then we have a set of integers which contains n, which is infinite, giving us an infinite integer.
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Yep, the set of integers is infinite. It is countably infinite (as is the set of all rational numbers). The irrationals and reals are not countable, although they are also infinite. So the set of reals is larger than the set of rational numbers, even though both are infinite.
Another way to look at it, in terms of countability, is that every integer is specifiable in arithmetic terms: namely the unique number of times that 1 is added to 0.
So if you were right, then you could tell me two unique numbers, that arithmetically compute two integers, whose ratio equals Pi. Right now, all you're telling me is that those two numbers are "infinity" and "infinity". But infinity is not a number in the sense that it is subject to the laws of arithmetic. So we're back where we started from.
That doesn't mean that you couldn't evaluate the limit (as phukraut points out). It just means that you can't express the limit as the ratio of integers. Rather it's always going to be an infinite sum of some kind, that you can compute as far out as you want.