Since $n\geq 4$ $n=5\:or\:6\:or\:7.....$
$1!+2!+3!+4!=33$
5!,6!,..............n! all these numbers end with 0.
$\therefore$$1!+2!+3!+4!+5!+..........n! = 33$ + (any number ending with 0)
= any number ending with 3.
$\Rightarrow\: $ It cannot be even, cannot be a square number.
$\therefore$ It is not a square of any positive integer.