Is this number irrational?
A formal justification of your informal proof can be achieved by noting that in the process of long division, the fact that you have a finite number of possible remainders guarantees that eventually a remainder will be repeated. That is, for any rational number its decimal expansion becomes periodic.
Yes, a number is rational if and only if its decimal representation is eventually periodic (including the possibility of a period $\overline 0$)
A formal proof for your specific number requires a formal definition. I assume that your number is $$\sum_{n=1}^\infty 10^{-\frac{n^2+n}2}.$$ Any decimal representation that has infinitely non-zero digits (which is the case for your number) and has blocks of zeroes of arbitrary size (which is also the case for your number) cannot be eventually periodic: Some late period must lie completely in a sufficiently big block of zeroes, hence the period must be all zeroes, contradicting the fact that some non-zero digit occurs further to the right.
A number with a similar expression can even be shown to be not only irrational, but in fact transcendent: $$\sum_{n=1}^\infty 10^{-n!}.$$