A series is monotonic if each term is larger (or smaller, but in only one direction) than the previous term. We can prove this by looking at the first two terms (n=2, 3) and then by looking at a general case (n and n+1). Let's start with the 2 and 3.
53!/6! > 52!/4! (Let's cross multiply)
53!•4! > 52!•6! (Let's divide both sides by 52! and 4!)
53 > 30
So we know it works for the first case. Let's look at the general case.
(n+51)!/(2n+2)! > (n+50)!/(2n)! (Let's cross multiply)
(n+1)!•(2n)! > (n+50)!•(2n+2)! (Let's divide both sides by (n+50)! and (2n)!)
(n+51) > (2n+2)•(2n+1)
This is not true for large values of n (switches to decreasing at n=3). So since it is increasing for some values and decreasing for others, we know that this sequence is not monotonic. If we instead started at n=3, then it would be monotonic.
