Infinite union of regular language Deciding if the infinite union of a set of regular languages is regular is undecidable.

By closure property of regular languages, regular language is not closed under infinite union so is the above problem undecidable?
Since every language is a countable union of regular languages, you're basically asking whether one can decide whether a given language is regular. Thus, for any reasonable representation, this task will be undecidable.

Consider the following representation of your problem, say a machine $M$ represents an infinite series of regular expressions if given input $i$, $M$ produces a regular expression $r^M_i$. You now ask whether the following language is decidable:

$L=\left\{\langle M\rangle \Big| \text{$\bigcup\limits_{i}L\left(r_i^M\right)$ is regular}\right\}$

If $M$ does not represent a series of regular expressions, i.e. on some input $i$, $M(i)$ is not a valid regular expression, then $\langle M\rangle\notin L$. $L$ is undecidable, since you can reduce the language $L_R=\{\langle M\rangle | \text{$L(M)$ is regular}\}$ to it ($L_R$ is trivially undecidable by Rice theorem). Your reduction will, given $M$, generate a machine $M'$ which on input $i$ will simulate $M$ on all inputs of length $\le i$ for $i$ steps, and according to the result, will generate a regular expression which corresponds to the union of all words that were accepted during this simulation. Clearly $\bigcup\limits_{i}L\left(r_i^{M'}\right)=L(M)$, and the validity of the reduction immediately follows.