# I have found an example where regular expression is not closed under concatenation. Where am I wrong?

$$a^n$$ is a regular expression. $$b^n$$ is a regular expression. their concatenation is $$a^nb^n$$ which is not a regular expression.

If $$n$$ is fixed then $$a^n$$ is just a single word and so is $$a^nb^n$$.
If by $$a^n$$ you mean the language $$\{a^n \mid n \ge 0\}$$ (whose corresponding regular expression is $$a^*$$) then the problem is that the variable $$n$$ in $$a^n$$ is independent of the variable with the same name in $$b^n$$.
The language obtained by concatenating $$A=\{a^n \mid n \ge 0\}$$ with $$B=\{b^n \mid n \ge 0\}$$ is $$\{xy \mid x \in A, y \in B\} = \{a^nb^m \mid n,m \ge 0\}$$ and the corresponding regular expression is $$a^*b^*$$. As you can see this is exactly the concatenation of $$a^*$$ and $$b^*$$.