We say, something can go to infinity such as $\mathrm{e}^n$ goes to infinity faster than, say, $1+\frac{1}{2}+\frac{1}{3}+\cdots$. But they are all infinity.

Seems like we only care about the speed they go to infinity, not never think about what they looks like when they are already infinity.

For example, $(2+\sqrt{2})^n$ is an irrational number for any finite $n$. But when $n \rightarrow \infty$, $(2+\sqrt{2})^n \rightarrow $ integer. It not converges to any perticular integer number, but the "limit" belongs to the category of "integer".

$$(2+\sqrt{2})^n=2^n+C_{n}^{1}{2^{n-1}\sqrt{2}}+C_{n}^{2}{2^{n-2}{\sqrt{2}}^2}+\cdots$$

All odd items are integers, let the sum of all odd items $A$; and even items are irrational numbers, let the sum $B$. so

$$(2+\sqrt{2})^n=A+B$$

Noticing that $(2-\sqrt{2})^n=A-B \rightarrow 0$, we can easily find that $A>B$，and most importantly $B \rightarrow A$. A big irrational number $B$, is more and more close to integer $A$.

$(2+\sqrt{2})^n$ is infinitely close to integer.

Is there any branch of math research the property of infinity itself? An infinite integer has infinite decimal numbers, what's the difference with the irrational number BTW, which also has infinite decimal numbers.